Lie algebra sp(10), type \(C^{1}_5\)
Semisimple complex Lie subalgebras

sp(10), type \(C^{1}_5\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.
Page generated by the calculator project.
Up to linear equivalence, there are total 119 semisimple subalgebras (including the full subalgebra). The subalgebras are ordered by rank, Dynkin indices of simple constituents and dimensions of simple constituents.
The upper index indicates the Dynkin index, the lower index indicates the rank of the subalgebra. Generation comments.
Computation time in seconds: 7845.16.
18936120030 total arithmetic operations performed = 18707885725 additions and 228234305 multiplications.
The base field over which the subalgebras were realized is: \(\displaystyle \mathbb Q\)
Number of root subalgebras other than the Cartan and full subalgebra: 34
Number of sl(2)'s: 23
Subalgebra \(A^{1}_1\) ↪ \(C^{1}_5\)
1 out of 119
Subalgebra type: \(\displaystyle A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: \(\displaystyle C^{1}_4\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_1\)
Basis of Cartan of centralizer: 4 vectors: (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{1}_1\) , \(\displaystyle A^{2}_1+A^{1}_1\) , \(\displaystyle A^{3}_1+A^{1}_1\) , \(\displaystyle A^{4}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{1}_1\) , \(\displaystyle A^{9}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+A^{1}_1\) , \(\displaystyle A^{11}_1+A^{1}_1\) , \(\displaystyle A^{12}_1+A^{1}_1\) , \(\displaystyle A^{20}_1+A^{1}_1\) , \(\displaystyle A^{35}_1+A^{1}_1\) , \(\displaystyle A^{36}_1+A^{1}_1\) , \(\displaystyle A^{84}_1+A^{1}_1\) , \(\displaystyle 3A^{1}_1\) , \(\displaystyle A^{2}_1+2A^{1}_1\) , \(\displaystyle 2A^{2}_1+A^{1}_1\) , \(\displaystyle A^{3}_1+2A^{1}_1\) , \(\displaystyle 2A^{4}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+2A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{4}_1+A^{1}_1\) , \(\displaystyle A^{9}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+2A^{1}_1\) , \(\displaystyle A^{10}_1+A^{2}_1+A^{1}_1\) , \(\displaystyle 2A^{10}_1+A^{1}_1\) , \(\displaystyle A^{11}_1+2A^{1}_1\) , \(\displaystyle A^{35}_1+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{1}_1\) , \(\displaystyle A^{2}_2+A^{1}_1\) , \(\displaystyle B^{2}_2+A^{1}_1\) , \(\displaystyle 4A^{1}_1\) , \(\displaystyle A^{2}_1+3A^{1}_1\) , \(\displaystyle 3A^{4}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) , \(\displaystyle A^{10}_1+3A^{1}_1\) , \(\displaystyle B^{1}_2+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{2}_1+A^{1}_1\) , \(\displaystyle B^{1}_2+A^{10}_1+A^{1}_1\) , \(\displaystyle A^{2}_2+2A^{1}_1\) , \(\displaystyle C^{1}_3+A^{1}_1\) , \(\displaystyle A^{2}_3+A^{1}_1\) , \(\displaystyle 5A^{1}_1\) , \(\displaystyle B^{1}_2+3A^{1}_1\) , \(\displaystyle 2B^{1}_2+A^{1}_1\) , \(\displaystyle C^{1}_3+2A^{1}_1\) , \(\displaystyle C^{1}_4+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 2, 2, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-25}\)
Positive simple generators: \(\displaystyle g_{25}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus 8V_{\omega_{1}}\oplus 36V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{-2\psi_{3}+4\psi_{4}}\oplus V_{\psi_{1}-\psi_{3}+2\psi_{4}}\oplus V_{\omega_{1}-\psi_{3}+2\psi_{4}}\oplus V_{2\psi_{1}}
\oplus V_{\omega_{1}+\psi_{1}}\oplus V_{2\omega_{1}}\oplus V_{-\psi_{2}+2\psi_{4}}\oplus V_{-\psi_{1}+\psi_{2}-\psi_{3}+2\psi_{4}}
\oplus V_{\psi_{1}-\psi_{2}-\psi_{3}+2\psi_{4}}\oplus V_{\psi_{2}-2\psi_{3}+2\psi_{4}}\oplus V_{\psi_{1}-\psi_{2}+\psi_{3}}
\oplus V_{\omega_{1}-\psi_{2}+\psi_{3}}\oplus V_{\psi_{2}}\oplus V_{\omega_{1}-\psi_{1}+\psi_{2}}\oplus V_{2\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}}
\oplus V_{\psi_{1}+\psi_{2}-\psi_{3}}\oplus V_{\omega_{1}+\psi_{2}-\psi_{3}}\oplus V_{-\psi_{1}-\psi_{3}+2\psi_{4}}\oplus V_{-2\psi_{2}+2\psi_{3}}
\oplus V_{-\psi_{1}+\psi_{3}}\oplus V_{\psi_{1}-2\psi_{2}+\psi_{3}}\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 4V_{0}\oplus V_{\omega_{1}-\psi_{1}}
\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{-\psi_{1}+2\psi_{2}-\psi_{3}}\oplus V_{\psi_{1}-\psi_{3}}\oplus V_{2\psi_{2}-2\psi_{3}}\oplus V_{\psi_{1}+\psi_{3}-2\psi_{4}}
\oplus V_{\omega_{1}+\psi_{3}-2\psi_{4}}\oplus V_{-\psi_{1}-\psi_{2}+\psi_{3}}\oplus V_{-2\psi_{1}+\psi_{2}}\oplus V_{-\psi_{2}}\oplus V_{-\psi_{1}+\psi_{2}-\psi_{3}}
\oplus V_{-\psi_{2}+2\psi_{3}-2\psi_{4}}\oplus V_{-\psi_{1}+\psi_{2}+\psi_{3}-2\psi_{4}}\oplus V_{\psi_{1}-\psi_{2}+\psi_{3}-2\psi_{4}}
\oplus V_{\psi_{2}-2\psi_{4}}\oplus V_{-2\psi_{1}}\oplus V_{-\psi_{1}+\psi_{3}-2\psi_{4}}\oplus V_{2\psi_{3}-4\psi_{4}}\)
Made total 280 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1\) ↪ \(C^{1}_5\)
2 out of 119
Subalgebra type: \(\displaystyle A^{2}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: \(\displaystyle C^{1}_3\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2\)
Basis of Cartan of centralizer: 4 vectors: (1, 0, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_1+A^{1}_1\) , \(\displaystyle 2A^{2}_1\) , \(\displaystyle A^{3}_1+A^{2}_1\) , \(\displaystyle A^{8}_1+A^{2}_1\) , \(\displaystyle A^{10}_1+A^{2}_1\) , \(\displaystyle A^{11}_1+A^{2}_1\) , \(\displaystyle A^{35}_1+A^{2}_1\) , \(\displaystyle A^{2}_1+2A^{1}_1\) , \(\displaystyle 2A^{2}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{2}_1\) , \(\displaystyle A^{10}_1+A^{2}_1+A^{1}_1\) , \(\displaystyle B^{1}_2+A^{2}_1\) , \(\displaystyle A^{2}_2+A^{2}_1\) , \(\displaystyle A^{2}_1+3A^{1}_1\) , \(\displaystyle B^{1}_2+A^{2}_1+A^{1}_1\) , \(\displaystyle C^{1}_3+A^{2}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 4, 4, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-24}\)
Positive simple generators: \(\displaystyle g_{24}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{1}}\oplus 12V_{\omega_{1}}\oplus 22V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{3}+2\psi_{4}}\oplus V_{\omega_{1}+\psi_{1}+\psi_{2}}\oplus V_{-2\psi_{3}+4\psi_{4}}
\oplus V_{\psi_{2}-\psi_{3}+2\psi_{4}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}+\psi_{3}}\oplus V_{2\psi_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+\psi_{1}+\psi_{2}-\psi_{3}}
\oplus V_{-\psi_{2}+2\psi_{4}}\oplus V_{\omega_{1}-\psi_{1}-\psi_{3}+2\psi_{4}}\oplus V_{\psi_{2}-2\psi_{3}+2\psi_{4}}\oplus V_{\psi_{3}}
\oplus V_{\omega_{1}-\psi_{1}+\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}}\oplus V_{2\psi_{2}-\psi_{3}}\oplus V_{\omega_{1}+\psi_{1}+\psi_{3}-2\psi_{4}}
\oplus V_{-\psi_{2}-\psi_{3}+2\psi_{4}}\oplus V_{-2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}-\psi_{1}-\psi_{2}+\psi_{3}}\oplus 4V_{0}
\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}+\psi_{2}-\psi_{3}}\oplus V_{2\psi_{2}-2\psi_{3}}\oplus V_{\psi_{2}+\psi_{3}-2\psi_{4}}
\oplus V_{-2\psi_{2}+\psi_{3}}\oplus V_{\omega_{1}-\psi_{1}-\psi_{2}}\oplus V_{-\psi_{3}}\oplus V_{-\psi_{2}+2\psi_{3}-2\psi_{4}}
\oplus V_{\omega_{1}-\psi_{1}+\psi_{3}-2\psi_{4}}\oplus V_{\psi_{2}-2\psi_{4}}\oplus V_{-2\psi_{2}}\oplus V_{-\psi_{2}+\psi_{3}-2\psi_{4}}
\oplus V_{2\psi_{3}-4\psi_{4}}\)
Made total 14342990418 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1\) ↪ \(C^{1}_5\)
3 out of 119
Subalgebra type: \(\displaystyle A^{3}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle B^{1}_2+A^{8}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{3}_1\)
Basis of Cartan of centralizer: 3 vectors: (1, 0, 0, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{3}_1+A^{1}_1\) , \(\displaystyle A^{3}_1+A^{2}_1\) , \(\displaystyle A^{8}_1+A^{3}_1\) , \(\displaystyle A^{9}_1+A^{3}_1\) , \(\displaystyle A^{10}_1+A^{3}_1\) , \(\displaystyle A^{10}_1+A^{3}_1\) , \(\displaystyle A^{18}_1+A^{3}_1\) , \(\displaystyle A^{3}_1+2A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{2}_1\) , \(\displaystyle A^{9}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+A^{8}_1+A^{3}_1\) , \(\displaystyle B^{1}_2+A^{3}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{8}_1+A^{3}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (2, 4, 6, 6, 3): 6
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-19}+g_{-24}\)
Positive simple generators: \(\displaystyle g_{24}+g_{19}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 6V_{2\omega_{1}}\oplus 12V_{\omega_{1}}\oplus 13V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}+\psi_{1}}
\oplus V_{-2\psi_{2}+4\psi_{3}}\oplus V_{2\psi_{3}}\oplus V_{\omega_{1}-\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{2}}\oplus V_{\omega_{1}+\psi_{2}}
\oplus 2V_{2\omega_{1}}\oplus V_{\omega_{1}-\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}-\psi_{1}+\psi_{2}}\oplus V_{\psi_{1}}
\oplus V_{2\omega_{1}-\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}+\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{2}+2\psi_{3}}
\oplus 3V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}-\psi_{2}}\oplus V_{2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+\psi_{2}-2\psi_{3}}
\oplus V_{-\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}-\psi_{1}+\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{2}}
\oplus V_{-2\psi_{3}}\oplus V_{2\psi_{2}-4\psi_{3}}\)
Made total 1235361456 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1\) ↪ \(C^{1}_5\)
4 out of 119
Subalgebra type: \(\displaystyle A^{4}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle 2A^{4}_1+A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{4}_1\)
Basis of Cartan of centralizer: 3 vectors: (1, 0, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{4}_1+A^{1}_1\) , \(\displaystyle 2A^{4}_1\) , \(\displaystyle A^{5}_1+A^{4}_1\) , \(\displaystyle A^{8}_1+A^{4}_1\) , \(\displaystyle A^{9}_1+A^{4}_1\) , \(\displaystyle 2A^{4}_1+A^{1}_1\) , \(\displaystyle 3A^{4}_1\) , \(\displaystyle A^{5}_1+2A^{4}_1\) , \(\displaystyle A^{8}_1+A^{4}_1+A^{1}_1\) , \(\displaystyle 3A^{4}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 6, 8, 4): 8
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-16}+g_{-24}\)
Positive simple generators: \(\displaystyle g_{24}+g_{16}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 10V_{2\omega_{1}}\oplus 8V_{\omega_{1}}\oplus 9V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi_{3}}\oplus V_{\omega_{1}+\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\psi_{1}+2\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{2}}
\oplus V_{2\omega_{1}+\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}+2\psi_{3}}\oplus V_{\omega_{1}-\psi_{2}+2\psi_{3}}
\oplus V_{\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}-\psi_{1}+\psi_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{2\omega_{1}+\psi_{1}-\psi_{2}}
\oplus V_{-\psi_{1}+\psi_{2}}\oplus 3V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\psi_{1}-\psi_{2}}\oplus V_{2\omega_{1}-\psi_{1}-\psi_{2}}
\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+\psi_{1}-2\psi_{3}}\oplus V_{-\psi_{1}-\psi_{2}}
\oplus V_{\omega_{1}-\psi_{1}-2\psi_{3}}\oplus V_{\omega_{1}-\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{3}}\)
Made total 2345117399 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{5}_1\) ↪ \(C^{1}_5\)
5 out of 119
Subalgebra type: \(\displaystyle A^{5}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle B^{4}_2\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{5}_1\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, 0, 0, 0), (0, 0, 1, 0, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{5}_1+A^{4}_1\) , \(\displaystyle A^{8}_1+A^{5}_1\) , \(\displaystyle A^{40}_1+A^{5}_1\) , \(\displaystyle A^{5}_1+2A^{4}_1\) , \(\displaystyle B^{4}_2+A^{5}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{5}_1\): (2, 4, 6, 8, 5): 10
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-5}+g_{-16}+g_{-24}\)
Positive simple generators: \(\displaystyle g_{24}+g_{16}+g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}10\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 15V_{2\omega_{1}}\oplus 10V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{2\omega_{1}+\psi_{2}}
\oplus V_{2\omega_{1}+\psi_{1}}\oplus V_{\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}-\psi_{1}+\psi_{2}}\oplus 3V_{2\omega_{1}}\oplus V_{2\omega_{1}+\psi_{1}-\psi_{2}}
\oplus V_{\psi_{2}}\oplus V_{\psi_{1}}\oplus V_{2\omega_{1}-\psi_{1}}\oplus V_{2\omega_{1}-\psi_{2}}\oplus V_{-\psi_{1}+\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}
\oplus V_{\psi_{1}-\psi_{2}}\oplus V_{2\omega_{1}-\psi_{1}-\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{-\psi_{1}}\oplus V_{-\psi_{2}}
\oplus V_{-\psi_{1}-\psi_{2}}\)
Made total 190206646 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1\) ↪ \(C^{1}_5\)
6 out of 119
Subalgebra type: \(\displaystyle A^{8}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle B^{1}_2+A^{3}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{8}_1\)
Basis of Cartan of centralizer: 3 vectors: (0, 0, 1, 0, 0), (1, 0, 0, 1, 0), (2, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{2}_1\) , \(\displaystyle A^{8}_1+A^{3}_1\) , \(\displaystyle A^{8}_1+A^{4}_1\) , \(\displaystyle A^{8}_1+A^{5}_1\) , \(\displaystyle A^{10}_1+A^{8}_1\) , \(\displaystyle A^{13}_1+A^{8}_1\) , \(\displaystyle A^{8}_1+2A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{2}_1\) , \(\displaystyle A^{8}_1+A^{4}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+A^{8}_1+A^{3}_1\) , \(\displaystyle B^{1}_2+A^{8}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{8}_1+A^{3}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-11}+g_{-17}\)
Positive simple generators: \(\displaystyle 2g_{17}+2g_{11}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{1}}\oplus 9V_{2\omega_{1}}\oplus 13V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\psi_{2}+4\psi_{3}}\oplus V_{2\psi_{2}+4\psi_{3}}\oplus V_{2\omega_{1}+\psi_{1}+\psi_{2}+2\psi_{3}}
\oplus V_{2\omega_{1}-\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{1}+\psi_{1}+2\psi_{3}}\oplus V_{2\omega_{1}-\psi_{1}+\psi_{2}+2\psi_{3}}
\oplus V_{4\omega_{1}}\oplus V_{2\psi_{1}}\oplus V_{2\omega_{1}}\oplus V_{\psi_{2}}\oplus V_{2\psi_{1}-\psi_{2}}\oplus V_{-2\psi_{1}+2\psi_{2}}
\oplus 3V_{0}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}+\psi_{1}-\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{1}+\psi_{2}}\oplus V_{-\psi_{2}}
\oplus V_{2\omega_{1}-\psi_{1}-2\psi_{3}}\oplus V_{2\omega_{1}+\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{1}}\oplus V_{2\omega_{1}-\psi_{1}-\psi_{2}-2\psi_{3}}
\oplus V_{4\omega_{1}-2\psi_{2}-4\psi_{3}}\oplus V_{-2\psi_{2}-4\psi_{3}}\)
Made total 9979170 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_1\) ↪ \(C^{1}_5\)
7 out of 119
Subalgebra type: \(\displaystyle A^{9}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{3}_1+A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{8}_1+A^{1}_1\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, 0, 1, 0), (2, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{9}_1+A^{1}_1\) , \(\displaystyle A^{9}_1+A^{3}_1\) , \(\displaystyle A^{9}_1+A^{4}_1\) , \(\displaystyle A^{9}_1+A^{3}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_1\): (4, 8, 10, 10, 5): 18
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-11}+g_{-17}+g_{-19}\)
Positive simple generators: \(\displaystyle g_{19}+2g_{17}+2g_{11}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{1}}\oplus 2V_{3\omega_{1}}\oplus 6V_{2\omega_{1}}\oplus 4V_{\omega_{1}}\oplus 6V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+\psi_{1}+2\psi_{2}}
\oplus V_{\omega_{1}+\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}}
\oplus 2V_{2\omega_{1}}\oplus 2V_{0}\oplus V_{\omega_{1}-\psi_{1}}\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{3\omega_{1}-\psi_{1}-2\psi_{2}}
\oplus V_{-2\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}-2\psi_{2}}\oplus V_{4\omega_{1}-2\psi_{1}-4\psi_{2}}
\oplus V_{-2\psi_{1}-4\psi_{2}}\)
Made total 5052616 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1\) ↪ \(C^{1}_5\)
8 out of 119
Subalgebra type: \(\displaystyle A^{10}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{3}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{8}_1\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 1, 0, 0), (2, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+A^{3}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (4, 8, 10, 12, 6): 20
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-11}+g_{-16}+g_{-17}\)
Positive simple generators: \(\displaystyle 2g_{17}+g_{16}+2g_{11}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{1}}\oplus 4V_{3\omega_{1}}\oplus 4V_{2\omega_{1}}\oplus 4V_{\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+4\psi_{2}}\oplus V_{3\omega_{1}+\psi_{1}+2\psi_{2}}\oplus V_{4\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}+2\psi_{2}}
\oplus V_{3\omega_{1}-\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{4\omega_{1}}\oplus V_{\omega_{1}-\psi_{1}+2\psi_{2}}
\oplus 2V_{2\omega_{1}}\oplus V_{3\omega_{1}+\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}-2\psi_{2}}
\oplus V_{3\omega_{1}-\psi_{1}-2\psi_{2}}\oplus V_{4\omega_{1}-4\psi_{2}}\oplus V_{\omega_{1}-\psi_{1}-2\psi_{2}}\oplus V_{-4\psi_{2}}\)
Made total 31058778 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1\) ↪ \(C^{1}_5\)
9 out of 119
Subalgebra type: \(\displaystyle A^{10}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle C^{1}_3\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2\)
Basis of Cartan of centralizer: 3 vectors: (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+A^{2}_1\) , \(\displaystyle A^{10}_1+A^{3}_1\) , \(\displaystyle A^{10}_1+A^{8}_1\) , \(\displaystyle 2A^{10}_1\) , \(\displaystyle A^{11}_1+A^{10}_1\) , \(\displaystyle A^{35}_1+A^{10}_1\) , \(\displaystyle A^{10}_1+2A^{1}_1\) , \(\displaystyle A^{10}_1+A^{2}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+A^{8}_1+A^{3}_1\) , \(\displaystyle 2A^{10}_1+A^{1}_1\) , \(\displaystyle B^{1}_2+A^{10}_1\) , \(\displaystyle A^{2}_2+A^{10}_1\) , \(\displaystyle A^{10}_1+3A^{1}_1\) , \(\displaystyle B^{1}_2+A^{10}_1+A^{1}_1\) , \(\displaystyle C^{1}_3+A^{10}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 6V_{3\omega_{1}}\oplus V_{2\omega_{1}}\oplus 21V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}-\psi_{2}+2\psi_{3}}\oplus V_{3\omega_{1}+\psi_{1}}\oplus V_{3\omega_{1}-\psi_{1}+\psi_{2}}
\oplus V_{3\omega_{1}+\psi_{1}-\psi_{2}}\oplus V_{-2\psi_{2}+4\psi_{3}}\oplus V_{\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{1}}
\oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}-\psi_{1}}\oplus V_{3\omega_{1}+\psi_{2}-2\psi_{3}}\oplus V_{-\psi_{1}+2\psi_{3}}\oplus V_{\psi_{1}-2\psi_{2}+2\psi_{3}}
\oplus V_{\psi_{2}}\oplus V_{2\psi_{1}-\psi_{2}}\oplus V_{-\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 3V_{0}\oplus V_{2\psi_{1}-2\psi_{2}}
\oplus V_{\psi_{1}+\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{1}+\psi_{2}}\oplus V_{-\psi_{2}}\oplus V_{-\psi_{1}+2\psi_{2}-2\psi_{3}}\oplus V_{\psi_{1}-2\psi_{3}}
\oplus V_{-2\psi_{1}}\oplus V_{-\psi_{1}+\psi_{2}-2\psi_{3}}\oplus V_{2\psi_{2}-4\psi_{3}}\)
Made total 1780 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{11}_1\) ↪ \(C^{1}_5\)
10 out of 119
Subalgebra type: \(\displaystyle A^{11}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle B^{1}_2\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{11}_1+A^{1}_1\) , \(\displaystyle A^{11}_1+A^{2}_1\) , \(\displaystyle A^{11}_1+A^{10}_1\) , \(\displaystyle A^{11}_1+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{11}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (6, 8, 10, 10, 5): 22
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-19}+g_{-23}\)
Positive simple generators: \(\displaystyle 4g_{23}+g_{19}+3g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 4V_{3\omega_{1}}\oplus 3V_{2\omega_{1}}\oplus 4V_{\omega_{1}}\oplus 10V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}-\psi_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+\psi_{1}}\oplus V_{4\omega_{1}}\oplus V_{-2\psi_{1}+4\psi_{2}}
\oplus V_{2\psi_{2}}\oplus V_{\omega_{1}-\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}}\oplus 3V_{2\omega_{1}}\oplus V_{3\omega_{1}-\psi_{1}}
\oplus V_{3\omega_{1}+\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}-\psi_{1}}\oplus V_{2\psi_{1}-2\psi_{2}}
\oplus V_{\omega_{1}+\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\)
Made total 829374 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{12}_1\) ↪ \(C^{1}_5\)
11 out of 119
Subalgebra type: \(\displaystyle A^{12}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 1, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{12}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{12}_1\): (6, 8, 10, 12, 6): 24
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-16}+g_{-23}\)
Positive simple generators: \(\displaystyle 4g_{23}+g_{16}+3g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/6\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}24\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 2V_{4\omega_{1}}\oplus 2V_{3\omega_{1}}\oplus 6V_{2\omega_{1}}\oplus 4V_{\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+2\psi_{2}}\oplus V_{4\omega_{1}+\psi_{1}}\oplus V_{4\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}+2\psi_{2}}
\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{2\omega_{1}+\psi_{1}}\oplus V_{4\omega_{1}-\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}+2\psi_{2}}
\oplus 2V_{2\omega_{1}}\oplus V_{2\omega_{1}-\psi_{1}}\oplus V_{3\omega_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}-2\psi_{2}}
\oplus V_{\omega_{1}-\psi_{1}-2\psi_{2}}\oplus V_{-4\psi_{2}}\)
Made total 52399050 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{13}_1\) ↪ \(C^{1}_5\)
12 out of 119
Subalgebra type: \(\displaystyle A^{13}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{8}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{3}_1\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{13}_1+A^{8}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{13}_1\): (6, 8, 10, 12, 7): 26
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-9}+g_{-19}+g_{-23}\)
Positive simple generators: \(\displaystyle 4g_{23}+g_{19}+g_{9}+3g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/13\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}26\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 10V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{4\omega_{1}+\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\psi}\oplus V_{4\omega_{1}-\psi}
\oplus 4V_{2\omega_{1}}\oplus V_{\psi}\oplus 2V_{2\omega_{1}-\psi}\oplus V_{0}\oplus V_{2\omega_{1}-2\psi}\oplus V_{-\psi}\)
Made total 54301561 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{18}_1\) ↪ \(C^{1}_5\)
13 out of 119
Subalgebra type: \(\displaystyle A^{18}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{3}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{8}_1\)
Basis of Cartan of centralizer: 1 vectors: (0, 2, 0, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{18}_1+A^{3}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{18}_1\): (6, 10, 14, 16, 8): 36
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-8}+g_{-10}+g_{-13}+g_{-15}\)
Positive simple generators: \(\displaystyle 2g_{15}+4g_{13}+3g_{10}+2g_{8}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/9\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}36\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 2V_{5\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 2V_{3\omega_{1}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+4\psi}\oplus V_{5\omega_{1}+2\psi}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{4\omega_{1}}
\oplus V_{\omega_{1}+2\psi}\oplus V_{5\omega_{1}-2\psi}\oplus 2V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi}\oplus V_{0}\oplus V_{4\omega_{1}-4\psi}
\oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\)
Made total 1042253 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{20}_1\) ↪ \(C^{1}_5\)
14 out of 119
Subalgebra type: \(\displaystyle A^{20}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, -1, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{20}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{20}_1\): (6, 12, 14, 16, 8): 40
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-2}+g_{-10}+g_{-16}\)
Positive simple generators: \(\displaystyle 4g_{16}+3g_{10}+3g_{2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/10\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}40\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 4V_{3\omega_{1}}\oplus 3V_{2\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}+2\psi_{1}}\oplus V_{3\omega_{1}+\psi_{1}+2\psi_{2}}\oplus V_{6\omega_{1}}\oplus V_{4\psi_{2}}\oplus V_{3\omega_{1}-\psi_{1}+2\psi_{2}}
\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{4\omega_{1}}\oplus V_{6\omega_{1}-2\psi_{1}}\oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}+\psi_{1}-2\psi_{2}}
\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{3\omega_{1}-\psi_{1}-2\psi_{2}}\oplus V_{-4\psi_{2}}\)
Made total 28767865 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{21}_1\) ↪ \(C^{1}_5\)
15 out of 119
Subalgebra type: \(\displaystyle A^{21}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (1, 0, -1, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{21}_1\): (6, 12, 14, 16, 9): 42
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-2}+g_{-5}+g_{-10}+g_{-16}\)
Positive simple generators: \(\displaystyle 4g_{16}+3g_{10}+g_{5}+3g_{2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/21\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}42\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{6\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 6V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}+2\psi}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}+\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{4\omega_{1}}\oplus V_{6\omega_{1}-2\psi}
\oplus V_{2\omega_{1}+\psi}\oplus V_{4\omega_{1}-\psi}\oplus 2V_{2\omega_{1}}\oplus V_{2\omega_{1}-\psi}\oplus V_{0}\oplus V_{2\omega_{1}-2\psi}\)
Made total 44412910 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{35}_1\) ↪ \(C^{1}_5\)
16 out of 119
Subalgebra type: \(\displaystyle A^{35}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle B^{1}_2\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{35}_1+A^{1}_1\) , \(\displaystyle A^{35}_1+A^{2}_1\) , \(\displaystyle A^{35}_1+A^{10}_1\) , \(\displaystyle A^{35}_1+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{35}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{35}_1\): (10, 16, 18, 18, 9): 70
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-19}\)
Positive simple generators: \(\displaystyle 9g_{19}+8g_{2}+5g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{6\omega_{1}}\oplus 4V_{5\omega_{1}}\oplus V_{2\omega_{1}}\oplus 10V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{10\omega_{1}}\oplus V_{5\omega_{1}-\psi_{1}+2\psi_{2}}\oplus V_{5\omega_{1}+\psi_{1}}\oplus V_{6\omega_{1}}\oplus V_{5\omega_{1}-\psi_{1}}
\oplus V_{5\omega_{1}+\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{2}}\oplus V_{2\psi_{1}}\oplus V_{2\omega_{1}}\oplus V_{-2\psi_{1}+2\psi_{2}}
\oplus 2V_{0}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\)
Made total 7217 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{36}_1\) ↪ \(C^{1}_5\)
17 out of 119
Subalgebra type: \(\displaystyle A^{36}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{36}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{36}_1\): (10, 16, 18, 20, 10): 72
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-13}+g_{-19}\)
Positive simple generators: \(\displaystyle 9g_{19}+g_{13}+8g_{2}+5g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/18\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}72\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus 2V_{6\omega_{1}}\oplus 2V_{5\omega_{1}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{10\omega_{1}}\oplus V_{5\omega_{1}+2\psi}\oplus 2V_{6\omega_{1}}\oplus V_{4\psi}\oplus V_{4\omega_{1}}\oplus V_{\omega_{1}+2\psi}\oplus V_{5\omega_{1}-2\psi}
\oplus 2V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\)
Made total 1614898 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{37}_1\) ↪ \(C^{1}_5\)
18 out of 119
Subalgebra type: \(\displaystyle A^{37}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{37}_1\): (10, 16, 18, 20, 11): 74
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-9}+g_{-19}\)
Positive simple generators: \(\displaystyle 9g_{19}+g_{9}+8g_{2}+5g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/37\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}74\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus 3V_{6\omega_{1}}\oplus 2V_{4\omega_{1}}\oplus 4V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{10\omega_{1}}\oplus V_{6\omega_{1}+\psi}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}+\psi}\oplus V_{6\omega_{1}-\psi}\oplus V_{2\omega_{1}+2\psi}
\oplus V_{4\omega_{1}-\psi}\oplus 2V_{2\omega_{1}}\oplus V_{0}\oplus V_{2\omega_{1}-2\psi}\)
Made total 19693568 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{40}_1\) ↪ \(C^{1}_5\)
19 out of 119
Subalgebra type: \(\displaystyle A^{40}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{5}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{4}_2\)
Basis of Cartan of centralizer: 1 vectors: (2, 0, -2, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{40}_1+A^{5}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{40}_1\): (8, 16, 20, 24, 12): 80
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-2}+g_{-4}+g_{-10}+g_{-12}\)
Positive simple generators: \(\displaystyle 6g_{12}+4g_{10}+6g_{4}+4g_{2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/20\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}80\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{8\omega_{1}+4\psi}\oplus V_{4\omega_{1}+4\psi}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\psi}\oplus V_{4\omega_{1}}\oplus V_{8\omega_{1}-4\psi}
\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{4\omega_{1}-4\psi}\oplus V_{-4\psi}\)
Made total 2361636 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{45}_1\) ↪ \(C^{1}_5\)
20 out of 119
Subalgebra type: \(\displaystyle A^{45}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{45}_1\): (10, 16, 22, 24, 13): 90
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-3}+5/2g_{-7}+g_{-9}+g_{-11}\)
Positive simple generators: \(\displaystyle 25/4g_{13}+11/2g_{11}+13/2g_{9}-15/2g_{8}+g_{7}+g_{6}+g_{5}+3g_{3}+5g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/45\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}90\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus 3V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus 3V_{2\omega_{1}}\)
Made total 5158931 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{84}_1\) ↪ \(C^{1}_5\)
21 out of 119
Subalgebra type: \(\displaystyle A^{84}_1\) (click on type for detailed printout).
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{84}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{84}_1\): (14, 24, 30, 32, 16): 168
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-13}\)
Positive simple generators: \(\displaystyle 16g_{13}+15g_{3}+12g_{2}+7g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/42\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}168\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus 2V_{7\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{7\omega_{1}+2\psi}\oplus V_{6\omega_{1}}\oplus V_{7\omega_{1}-2\psi}\oplus V_{4\psi}\oplus V_{2\omega_{1}}
\oplus V_{0}\oplus V_{-4\psi}\)
Made total 21932 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{85}_1\) ↪ \(C^{1}_5\)
22 out of 119
Subalgebra type: \(\displaystyle A^{85}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{85}_1\): (14, 24, 30, 32, 17): 170
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-5}+g_{-13}\)
Positive simple generators: \(\displaystyle 16g_{13}+g_{5}+15g_{3}+12g_{2}+7g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/85\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}170\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus 2V_{6\omega_{1}}\oplus 2V_{2\omega_{1}}\)
Made total 2630405 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{165}_1\) ↪ \(C^{1}_5\)
23 out of 119
Subalgebra type: \(\displaystyle A^{165}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{165}_1\): (18, 32, 42, 48, 25): 330
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle 25g_{5}+24g_{4}+21g_{3}+16g_{2}+9g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/165\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}330\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{18\omega_{1}}\oplus V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{1}}\)
Made total 56496 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{1}_1\) ↪ \(C^{1}_5\)
24 out of 119
Subalgebra type: \(\displaystyle 2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) .
Centralizer: \(\displaystyle C^{1}_3\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2\)
Basis of Cartan of centralizer: 3 vectors: (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle 3A^{1}_1\) , \(\displaystyle A^{2}_1+2A^{1}_1\) , \(\displaystyle A^{3}_1+2A^{1}_1\) , \(\displaystyle A^{8}_1+2A^{1}_1\) , \(\displaystyle A^{10}_1+2A^{1}_1\) , \(\displaystyle A^{11}_1+2A^{1}_1\) , \(\displaystyle A^{35}_1+2A^{1}_1\) , \(\displaystyle 4A^{1}_1\) , \(\displaystyle A^{2}_1+3A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) , \(\displaystyle A^{10}_1+3A^{1}_1\) , \(\displaystyle B^{1}_2+2A^{1}_1\) , \(\displaystyle A^{2}_2+2A^{1}_1\) , \(\displaystyle 5A^{1}_1\) , \(\displaystyle B^{1}_2+3A^{1}_1\) , \(\displaystyle C^{1}_3+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 2, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 2, 2, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{-23}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{23}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 6V_{\omega_{2}}\oplus 6V_{\omega_{1}}\oplus 21V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{-2\psi_{2}+4\psi_{3}}\oplus V_{\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{\omega_{2}-\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}-\psi_{2}+2\psi_{3}}
\oplus V_{2\psi_{1}}\oplus V_{\omega_{2}+\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}
\oplus V_{-\psi_{1}+2\psi_{3}}\oplus V_{\psi_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{\psi_{2}}\oplus V_{\omega_{2}-\psi_{1}+\psi_{2}}\oplus V_{\omega_{1}-\psi_{1}+\psi_{2}}
\oplus V_{2\psi_{1}-\psi_{2}}\oplus V_{\omega_{2}+\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}}\oplus V_{-\psi_{1}-\psi_{2}+2\psi_{3}}
\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 3V_{0}\oplus V_{\omega_{2}-\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{\psi_{1}+\psi_{2}-2\psi_{3}}
\oplus V_{\omega_{2}+\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{1}+\psi_{2}}\oplus V_{-\psi_{2}}
\oplus V_{-\psi_{1}+2\psi_{2}-2\psi_{3}}\oplus V_{\psi_{1}-2\psi_{3}}\oplus V_{-2\psi_{1}}\oplus V_{-\psi_{1}+\psi_{2}-2\psi_{3}}
\oplus V_{2\psi_{2}-4\psi_{3}}\)
Made total 369 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1+A^{1}_1\) ↪ \(C^{1}_5\)
25 out of 119
Subalgebra type: \(\displaystyle A^{2}_1+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) .
Centralizer: \(\displaystyle B^{1}_2\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)
Basis of Cartan of centralizer: 3 vectors: (1, 0, 0, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_1+2A^{1}_1\) , \(\displaystyle 2A^{2}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+A^{2}_1+A^{1}_1\) , \(\displaystyle A^{2}_1+3A^{1}_1\) , \(\displaystyle B^{1}_2+A^{2}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 4, 4, 2): 4, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{-19}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{19}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus 4V_{\omega_{2}}\oplus 8V_{\omega_{1}}\oplus 11V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\psi_{1}+\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}}
\oplus V_{-2\psi_{2}+4\psi_{3}}\oplus V_{2\psi_{3}}\oplus V_{\omega_{2}-\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{2}}\oplus V_{\omega_{2}+\psi_{2}}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}-\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}-\psi_{1}+\psi_{2}}
\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}+\psi_{2}-2\psi_{3}}
\oplus V_{-2\psi_{2}+2\psi_{3}}\oplus 3V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{2}-\psi_{2}}\oplus V_{2\psi_{2}-2\psi_{3}}
\oplus V_{\omega_{2}+\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}-\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}-\psi_{1}+\psi_{2}-2\psi_{3}}
\oplus V_{-2\psi_{2}}\oplus V_{-2\psi_{3}}\oplus V_{2\psi_{2}-4\psi_{3}}\)
Made total 117162364 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{2}_1\) ↪ \(C^{1}_5\)
26 out of 119
Subalgebra type: \(\displaystyle 2A^{2}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{2}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 3 vectors: (1, 0, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{2}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 4, 4, 2): 4, \(\displaystyle A^{2}_1\): (0, 0, 2, 4, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{-16}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{16}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0\\ 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0\\ 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 5V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi_{3}}\oplus V_{\omega_{2}+\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\psi_{1}+2\psi_{3}}\oplus V_{2\omega_{2}+2\psi_{2}}
\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}+2\psi_{3}}
\oplus V_{\omega_{2}-\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}
\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}}\oplus 3V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}}
\oplus V_{2\omega_{2}-2\psi_{2}}\oplus V_{\omega_{2}+\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+\psi_{1}-2\psi_{3}}\oplus V_{\omega_{1}-\psi_{1}-2\psi_{3}}
\oplus V_{\omega_{2}-\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{3}}\)
Made total 147792114 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1+A^{1}_1\) ↪ \(C^{1}_5\)
27 out of 119
Subalgebra type: \(\displaystyle A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) .
Centralizer: \(\displaystyle A^{8}_1+A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{3}_1+A^{1}_1\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, 0, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{3}_1+2A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{9}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (2, 4, 6, 6, 3): 6, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-19}+g_{-24}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle g_{24}+g_{19}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus 3V_{\omega_{1}+\omega_{2}}\oplus 6V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 6V_{\omega_{1}}\oplus 6V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{2}+2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{2}}
\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}}\oplus V_{2\omega_{1}+\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{2}}
\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}}\oplus V_{2\omega_{1}-\psi_{1}}
\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}-2\psi_{2}}\oplus V_{-\psi_{1}}\oplus V_{\omega_{2}-2\psi_{2}}
\oplus V_{\omega_{1}-2\psi_{2}}\oplus V_{\omega_{1}-\psi_{1}-2\psi_{2}}\oplus V_{-4\psi_{2}}\)
Made total 1299913 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1+A^{2}_1\) ↪ \(C^{1}_5\)
28 out of 119
Subalgebra type: \(\displaystyle A^{3}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1\) .
Centralizer: \(\displaystyle A^{8}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{3}_1\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, 0, 0, 0), (0, 0, 0, 1, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+A^{2}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (2, 4, 6, 6, 3): 6, \(\displaystyle A^{2}_1\): (0, 0, 0, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-19}+g_{-24}\), \(\displaystyle g_{-9}\)
Positive simple generators: \(\displaystyle g_{24}+g_{19}\), \(\displaystyle g_{9}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & 0\\ 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & 0\\ 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{2}}\oplus 6V_{\omega_{1}+\omega_{2}}\oplus 6V_{2\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{2}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{2}}
\oplus V_{2\omega_{1}+\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}}
\oplus V_{\psi_{1}}\oplus V_{2\omega_{1}-\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}
\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}}\oplus V_{2\omega_{2}-2\psi_{2}}\oplus V_{-\psi_{1}}\)
Made total 7274345 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{4}_1+A^{1}_1\) ↪ \(C^{1}_5\)
29 out of 119
Subalgebra type: \(\displaystyle A^{4}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{4}_1\) .
Centralizer: \(\displaystyle 2A^{4}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{4}_1+A^{1}_1\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, 0, 0, 0), (0, 0, 1, 0, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{4}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{4}_1+A^{1}_1\) , \(\displaystyle 3A^{4}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 6, 8, 4): 8, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-16}+g_{-24}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}+g_{16}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus 10V_{2\omega_{1}}\oplus 6V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{2}}
\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}}\oplus V_{\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}-\psi_{1}+\psi_{2}}\oplus V_{2\omega_{2}}
\oplus 2V_{2\omega_{1}}\oplus V_{2\omega_{1}+\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{2}}
\oplus V_{-\psi_{1}+\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\psi_{1}-\psi_{2}}\oplus V_{2\omega_{1}-\psi_{1}-\psi_{2}}
\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{-\psi_{1}-\psi_{2}}\)
Made total 1533158 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{4}_1\) ↪ \(C^{1}_5\)
30 out of 119
Subalgebra type: \(\displaystyle 2A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{4}_1\) .
Centralizer: \(\displaystyle A^{4}_1+A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{4}_1\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, 1, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{4}_1+A^{1}_1\) , \(\displaystyle 3A^{4}_1\) , \(\displaystyle A^{5}_1+2A^{4}_1\) , \(\displaystyle 3A^{4}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 6, 8, 4): 8, \(\displaystyle A^{4}_1\): (2, 4, 2, 0, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle -g_{-6}+g_{-7}\)
Positive simple generators: \(\displaystyle g_{21}+g_{20}\), \(\displaystyle g_{7}-g_{6}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0\\ 0 & 1/2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0\\ 0 & 8\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 6V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\omega_{2}+2\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}+2\psi_{2}}\oplus V_{4\psi_{2}}\oplus V_{2\omega_{1}+2\omega_{2}}
\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi_{1}}
\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}}
\oplus V_{-4\psi_{2}}\)
Made total 3764573 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{5}_1+A^{4}_1\) ↪ \(C^{1}_5\)
31 out of 119
Subalgebra type: \(\displaystyle A^{5}_1+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{5}_1\) .
Centralizer: \(\displaystyle A^{4}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{4}_1+A^{1}_1\)
Basis of Cartan of centralizer: 1 vectors: (1, 0, 0, 1, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{5}_1+2A^{4}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{5}_1\): (2, 4, 6, 8, 5): 10, \(\displaystyle A^{4}_1\): (2, 4, 4, 2, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-17}+g_{-18}+g_{-19}\), \(\displaystyle -g_{-10}+g_{-11}\)
Positive simple generators: \(\displaystyle g_{19}+g_{18}+g_{17}\), \(\displaystyle g_{11}-g_{10}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/5 & 0\\ 0 & 1/2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}10 & 0\\ 0 & 8\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{2}}
\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\omega_{2}+2\psi}\oplus V_{2\omega_{1}+\omega_{2}+\psi}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\psi}\oplus V_{\omega_{2}+\psi}
\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}-\psi}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi}\oplus V_{0}\oplus V_{\omega_{2}-\psi}
\oplus V_{-2\psi}\)
Made total 5105422 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{1}_1\) ↪ \(C^{1}_5\)
32 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) .
Centralizer: \(\displaystyle A^{3}_1+A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{8}_1+A^{1}_1\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, 0, 1, 0), (2, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+2A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{4}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-11}+g_{-17}\), \(\displaystyle g_{-19}\)
Positive simple generators: \(\displaystyle 2g_{17}+2g_{11}\), \(\displaystyle g_{19}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 5V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 6V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\psi_{1}+2\psi_{2}}
\oplus V_{2\omega_{1}+2\psi_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\psi_{1}}\oplus V_{\omega_{2}+\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}
\oplus 2V_{0}\oplus V_{\omega_{2}-\psi_{1}}\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}}
\oplus V_{2\omega_{1}-2\psi_{1}-2\psi_{2}}\oplus V_{4\omega_{1}-2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-4\psi_{2}}\)
Made total 932 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{2}_1\) ↪ \(C^{1}_5\)
33 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) .
Centralizer: \(\displaystyle A^{3}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{8}_1\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 1, 0, 0), (2, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+A^{2}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{2}_1\): (0, 0, 2, 4, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-11}+g_{-17}\), \(\displaystyle g_{-16}\)
Positive simple generators: \(\displaystyle 2g_{17}+2g_{11}\), \(\displaystyle g_{16}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\ 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\ 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{1}}\oplus 4V_{2\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\psi_{1}+2\psi_{2}}\oplus V_{4\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-\psi_{1}+2\psi_{2}}
\oplus V_{2\omega_{2}+2\psi_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}+\psi_{1}-2\psi_{2}}
\oplus 2V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{2\omega_{1}+\omega_{2}-\psi_{1}-2\psi_{2}}\oplus V_{4\omega_{1}-4\psi_{2}}
\oplus V_{-4\psi_{2}}\)
Made total 7488844 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{3}_1\) ↪ \(C^{1}_5\)
34 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) .
Centralizer: \(\displaystyle B^{1}_2\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+A^{2}_1\) , \(\displaystyle A^{10}_1+A^{8}_1+A^{3}_1\) , \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{8}_1+A^{3}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{3}_1\): (2, 0, 2, 2, 1): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-6}+g_{-21}\), \(\displaystyle g_{-1}+g_{-19}\)
Positive simple generators: \(\displaystyle 2g_{21}+2g_{6}\), \(\displaystyle g_{19}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\ 0 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\ 0 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus 4V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 10V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}-\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\psi_{1}}
\oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{2}}\oplus V_{2\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}-\psi_{1}}
\oplus V_{2\omega_{1}+\omega_{2}+\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}}
\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\)
Made total 422471 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{4}_1\) ↪ \(C^{1}_5\)
35 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{4}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{4}_1\): (2, 0, 2, 4, 2): 8
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-10}+g_{-18}\), \(\displaystyle g_{-1}+g_{-13}+g_{-19}\)
Positive simple generators: \(\displaystyle 2g_{18}+2g_{10}\), \(\displaystyle g_{19}+g_{13}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\ 0 & 1/2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\ 0 & 8\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}
\oplus 2V_{\omega_{2}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi}\oplus V_{4\psi}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{\omega_{2}+2\psi}
\oplus 2V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi}\oplus V_{0}\oplus V_{\omega_{2}-2\psi}\oplus V_{-4\psi}\)
Made total 4731438 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{5}_1\) ↪ \(C^{1}_5\)
36 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{5}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 1, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{5}_1\): (2, 0, 2, 4, 3): 10
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-14}+g_{-15}\), \(\displaystyle g_{-1}+g_{-5}+g_{-16}\)
Positive simple generators: \(\displaystyle 2g_{15}+2g_{14}\), \(\displaystyle g_{16}+g_{5}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0\\ 0 & 2/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0\\ 0 & 10\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus 2V_{2\omega_{1}+2\omega_{2}}\oplus 4V_{2\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}+\psi}\oplus V_{2\omega_{2}+2\psi}\oplus V_{2\omega_{1}+\psi}\oplus V_{2\omega_{1}+2\omega_{2}-\psi}
\oplus 2V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}-\psi}\oplus V_{0}\oplus V_{2\omega_{2}-2\psi}\)
Made total 48944976 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_1+A^{1}_1\) ↪ \(C^{1}_5\)
37 out of 119
Subalgebra type: \(\displaystyle A^{9}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1\) .
Centralizer: \(\displaystyle A^{3}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{8}_1\)
Basis of Cartan of centralizer: 1 vectors: (2, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{9}_1+A^{3}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_1\): (4, 8, 10, 10, 5): 18, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-11}+g_{-17}+g_{-19}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle g_{19}+2g_{17}+2g_{11}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}
\oplus 2V_{\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+4\psi}\oplus V_{2\omega_{1}+\omega_{2}+2\psi}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{4\omega_{1}}\oplus V_{\omega_{1}+2\psi}
\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi}\oplus V_{3\omega_{1}-2\psi}
\oplus V_{0}\oplus V_{4\omega_{1}-4\psi}\oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\)
Made total 709 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_1+A^{3}_1\) ↪ \(C^{1}_5\)
38 out of 119
Subalgebra type: \(\displaystyle A^{9}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{9}_1+A^{3}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_1\): (4, 8, 10, 10, 5): 18, \(\displaystyle A^{3}_1\): (2, 0, 0, 2, 1): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-10}+g_{-18}+g_{-19}\), \(\displaystyle g_{-1}+g_{-13}\)
Positive simple generators: \(\displaystyle g_{19}+2g_{18}+2g_{10}\), \(\displaystyle g_{13}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & 0\\ 0 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & 0\\ 0 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi}\oplus V_{4\psi}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+2\psi}
\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi}\oplus V_{0}\oplus V_{\omega_{1}-2\psi}
\oplus V_{-4\psi}\)
Made total 682383 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_1+A^{4}_1\) ↪ \(C^{1}_5\)
39 out of 119
Subalgebra type: \(\displaystyle A^{9}_1+A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_1\): (4, 8, 10, 10, 5): 18, \(\displaystyle A^{4}_1\): (2, 0, 0, 2, 2): 8
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-14}+g_{-15}+g_{-19}\), \(\displaystyle g_{-1}+g_{-5}+g_{-13}\)
Positive simple generators: \(\displaystyle g_{19}+2g_{15}+2g_{14}\), \(\displaystyle g_{13}+g_{5}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & 0\\ 0 & 1/2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & 0\\ 0 & 8\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}
\oplus 3V_{2\omega_{1}}\)
Made total 6256102 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{1}_1\) ↪ \(C^{1}_5\)
40 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) .
Centralizer: \(\displaystyle B^{1}_2\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+2A^{1}_1\) , \(\displaystyle A^{10}_1+A^{2}_1+A^{1}_1\) , \(\displaystyle 2A^{10}_1+A^{1}_1\) , \(\displaystyle A^{10}_1+3A^{1}_1\) , \(\displaystyle B^{1}_2+A^{10}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-19}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle g_{19}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus 4V_{3\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{\omega_{2}}
\oplus 10V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}-\psi_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+\psi_{1}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}}
\oplus V_{2\psi_{2}}\oplus V_{\omega_{2}-\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}}\oplus V_{\omega_{2}+\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}
\oplus V_{3\omega_{1}-\psi_{1}}\oplus V_{3\omega_{1}+\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{2}-\psi_{1}}
\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}+\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\)
Made total 458 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{2}_1\) ↪ \(C^{1}_5\)
41 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 1, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+A^{2}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{2}_1\): (0, 0, 2, 4, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-16}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle g_{16}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\ 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\ 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus 3V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{\omega_{2}}
\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+\omega_{2}+\psi_{1}}\oplus V_{4\psi_{2}}\oplus V_{\omega_{2}+\psi_{1}+2\psi_{2}}
\oplus V_{2\omega_{2}+2\psi_{1}}\oplus V_{3\omega_{1}+\omega_{2}-\psi_{1}}\oplus V_{\omega_{2}-\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{2}}
\oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{\omega_{2}+\psi_{1}-2\psi_{2}}
\oplus V_{\omega_{2}-\psi_{1}-2\psi_{2}}\oplus V_{-4\psi_{2}}\)
Made total 6291863 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{3}_1\) ↪ \(C^{1}_5\)
42 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 1, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (4, 8, 10, 12, 6): 20, \(\displaystyle A^{3}_1\): (2, 0, 0, 0, 1): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-14}+g_{-15}+g_{-16}\), \(\displaystyle g_{-1}+g_{-5}\)
Positive simple generators: \(\displaystyle g_{16}+2g_{15}+2g_{14}\), \(\displaystyle g_{5}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\ 0 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\ 0 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 4V_{2\omega_{1}}
\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{1}+\omega_{2}+\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{\omega_{1}+\omega_{2}+\psi}
\oplus V_{3\omega_{1}+\omega_{2}-\psi}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi}\oplus V_{0}\oplus V_{2\omega_{1}-2\psi}\)
Made total 8196389 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{3}_1\) ↪ \(C^{1}_5\)
43 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) .
Centralizer: \(\displaystyle A^{8}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{3}_1\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+A^{8}_1+A^{3}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{3}_1\): (0, 0, 2, 4, 3): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-9}+g_{-19}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle g_{19}+g_{9}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\ 0 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\ 0 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 3V_{3\omega_{1}+\omega_{2}}\oplus 6V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}+\psi}\oplus V_{2\omega_{2}+2\psi}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}+\psi}
\oplus V_{3\omega_{1}+\omega_{2}-\psi}\oplus 2V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\psi}\oplus V_{2\omega_{2}-\psi}\oplus V_{0}\oplus V_{2\omega_{2}-2\psi}
\oplus V_{-\psi}\)
Made total 7172134 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{8}_1\) ↪ \(C^{1}_5\)
44 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) .
Centralizer: \(\displaystyle A^{3}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{8}_1\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 2, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+A^{8}_1+A^{3}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{8}_1\): (0, 0, 4, 8, 4): 16
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-4}+g_{-12}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle 2g_{12}+2g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\ 0 & 1/4\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\ 0 & 16\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+2\omega_{2}}\oplus 3V_{4\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{2}+4\psi}\oplus V_{3\omega_{1}+2\omega_{2}+2\psi}\oplus V_{6\omega_{1}}\oplus V_{4\psi}\oplus V_{4\omega_{2}}\oplus V_{3\omega_{1}+2\omega_{2}-2\psi}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{4\omega_{2}-4\psi}\oplus V_{-4\psi}\)
Made total 206648 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{10}_1\) ↪ \(C^{1}_5\)
45 out of 119
Subalgebra type: \(\displaystyle 2A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle 2A^{10}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{10}_1\): (0, 0, 6, 8, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-3}+g_{-13}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle 4g_{13}+3g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0\\ 0 & 1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0\\ 0 & 20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{2}}\oplus V_{3\omega_{1}+3\omega_{2}}\oplus V_{6\omega_{1}}\oplus 2V_{3\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus V_{2\omega_{2}}
\oplus V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{2}}\oplus V_{3\omega_{1}+3\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{2}+2\psi}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{3\omega_{2}-2\psi}\oplus V_{3\omega_{1}-2\psi}\oplus V_{0}\oplus V_{-4\psi}\)
Made total 2816 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{11}_1+A^{1}_1\) ↪ \(C^{1}_5\)
46 out of 119
Subalgebra type: \(\displaystyle A^{11}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{11}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{11}_1+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (6, 8, 10, 10, 5): 22, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-19}+g_{-23}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle 4g_{23}+g_{19}+3g_{1}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{3\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus 3V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{\omega_{2}+2\psi}
\oplus V_{\omega_{1}+2\psi}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi}\oplus V_{0}
\oplus V_{\omega_{2}-2\psi}\oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\)
Made total 703 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{11}_1+A^{2}_1\) ↪ \(C^{1}_5\)
47 out of 119
Subalgebra type: \(\displaystyle A^{11}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{11}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (6, 8, 10, 10, 5): 22, \(\displaystyle A^{2}_1\): (0, 0, 0, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-19}+g_{-23}\), \(\displaystyle g_{-9}\)
Positive simple generators: \(\displaystyle 4g_{23}+g_{19}+3g_{1}\), \(\displaystyle g_{9}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11 & 0\\ 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22 & 0\\ 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus 3V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}
\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}+\psi}\oplus V_{2\omega_{2}+2\psi}\oplus V_{4\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}+\psi}
\oplus V_{3\omega_{1}+\omega_{2}-\psi}\oplus V_{2\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi}\oplus V_{0}\oplus V_{2\omega_{2}-2\psi}\)
Made total 1282769 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{11}_1+A^{10}_1\) ↪ \(C^{1}_5\)
48 out of 119
Subalgebra type: \(\displaystyle A^{11}_1+A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{11}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (6, 8, 10, 10, 5): 22, \(\displaystyle A^{10}_1\): (0, 0, 0, 6, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-19}+g_{-23}\), \(\displaystyle g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{23}+g_{19}+3g_{1}\), \(\displaystyle 4g_{5}+3g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11 & 0\\ 0 & 1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22 & 0\\ 0 & 20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{2}}\oplus V_{3\omega_{1}+3\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{\omega_{1}+3\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}
\oplus 3V_{2\omega_{1}}\)
Made total 4368 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{12}_1+A^{1}_1\) ↪ \(C^{1}_5\)
49 out of 119
Subalgebra type: \(\displaystyle A^{12}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{12}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 1, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{12}_1\): (6, 8, 10, 12, 6): 24, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-16}+g_{-23}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{23}+g_{16}+3g_{1}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/6 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}24 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus 2V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 6V_{2\omega_{1}}
\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{4\omega_{1}+\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}+\psi}
\oplus V_{2\omega_{1}+\psi}\oplus V_{4\omega_{1}-\psi}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi}
\oplus V_{2\omega_{1}-\psi}\oplus V_{0}\oplus V_{2\omega_{1}-2\psi}\)
Made total 94415 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{13}_1+A^{8}_1\) ↪ \(C^{1}_5\)
50 out of 119
Subalgebra type: \(\displaystyle A^{13}_1+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{13}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{13}_1\): (6, 8, 10, 12, 7): 26, \(\displaystyle A^{8}_1\): (0, 4, 4, 4, 0): 16
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-6}+g_{-13}+g_{-15}+g_{-19}\), \(\displaystyle -g_{-4}+g_{-7}\)
Positive simple generators: \(\displaystyle 4g_{19}+g_{15}+g_{13}+3g_{6}\), \(\displaystyle 2g_{7}-2g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/13 & 0\\ 0 & 1/4\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}26 & 0\\ 0 & 16\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+4\omega_{2}}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{2}}
\oplus 2V_{2\omega_{1}}\)
Made total 7971893 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{18}_1+A^{3}_1\) ↪ \(C^{1}_5\)
51 out of 119
Subalgebra type: \(\displaystyle A^{18}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{18}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{18}_1\): (6, 10, 14, 16, 8): 36, \(\displaystyle A^{3}_1\): (0, 2, 0, 0, 1): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-10}+g_{-11}+g_{-12}+g_{-13}\), \(\displaystyle g_{-2}+g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{13}+2g_{12}+2g_{11}+3g_{10}\), \(\displaystyle g_{5}+g_{2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/9 & 0\\ 0 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}36 & 0\\ 0 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}
\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\)
Made total 1462843 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{20}_1+A^{1}_1\) ↪ \(C^{1}_5\)
52 out of 119
Subalgebra type: \(\displaystyle A^{20}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{20}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (1, 0, -1, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{20}_1\): (6, 12, 14, 16, 8): 40, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-2}+g_{-10}+g_{-16}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{16}+3g_{10}+3g_{2}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/10 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}40 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}+2\psi}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}+\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{4\omega_{1}}
\oplus V_{6\omega_{1}-2\psi}\oplus V_{3\omega_{1}+\omega_{2}-\psi}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{2\omega_{1}-2\psi}\)
Made total 95486 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{35}_1+A^{1}_1\) ↪ \(C^{1}_5\)
53 out of 119
Subalgebra type: \(\displaystyle A^{35}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{35}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{35}_1+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{35}_1\): (10, 16, 18, 18, 9): 70, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-19}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle 9g_{19}+8g_{2}+5g_{1}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus 2V_{5\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}
\oplus 2V_{\omega_{2}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{10\omega_{1}}\oplus V_{5\omega_{1}+2\psi}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{4\psi}\oplus V_{\omega_{2}+2\psi}
\oplus V_{5\omega_{1}-2\psi}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{2}-2\psi}\oplus V_{-4\psi}\)
Made total 541 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{35}_1+A^{2}_1\) ↪ \(C^{1}_5\)
54 out of 119
Subalgebra type: \(\displaystyle A^{35}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{35}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{35}_1\): (10, 16, 18, 18, 9): 70, \(\displaystyle A^{2}_1\): (0, 0, 0, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-19}\), \(\displaystyle g_{-9}\)
Positive simple generators: \(\displaystyle 9g_{19}+8g_{2}+5g_{1}\), \(\displaystyle g_{9}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35 & 0\\ 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70 & 0\\ 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus 2V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus 3V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{10\omega_{1}}\oplus V_{5\omega_{1}+\omega_{2}+\psi}\oplus V_{6\omega_{1}}\oplus V_{5\omega_{1}+\omega_{2}-\psi}\oplus V_{2\omega_{2}+2\psi}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{2\omega_{2}-2\psi}\)
Made total 1051879 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{35}_1+A^{10}_1\) ↪ \(C^{1}_5\)
55 out of 119
Subalgebra type: \(\displaystyle A^{35}_1+A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{35}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{35}_1\): (10, 16, 18, 18, 9): 70, \(\displaystyle A^{10}_1\): (0, 0, 0, 6, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-19}\), \(\displaystyle g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle 9g_{19}+8g_{2}+5g_{1}\), \(\displaystyle 4g_{5}+3g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35 & 0\\ 0 & 1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70 & 0\\ 0 & 20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{5\omega_{1}+3\omega_{2}}\oplus V_{6\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 3340 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{36}_1+A^{1}_1\) ↪ \(C^{1}_5\)
56 out of 119
Subalgebra type: \(\displaystyle A^{36}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{36}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{36}_1\): (10, 16, 18, 20, 10): 72, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-13}+g_{-19}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 9g_{19}+g_{13}+8g_{2}+5g_{1}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/18 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}72 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus 2V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus 2V_{2\omega_{1}}\)
Made total 786 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{40}_1+A^{5}_1\) ↪ \(C^{1}_5\)
57 out of 119
Subalgebra type: \(\displaystyle A^{40}_1+A^{5}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{40}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{40}_1\): (8, 16, 20, 24, 12): 80, \(\displaystyle A^{5}_1\): (2, 0, 2, 0, 1): 10
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-6}+g_{-7}+g_{-8}+g_{-9}\), \(\displaystyle g_{-1}+g_{-3}+g_{-5}\)
Positive simple generators: \(\displaystyle 6g_{9}+6g_{8}+4g_{7}+4g_{6}\), \(\displaystyle g_{5}+g_{3}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/20 & 0\\ 0 & 2/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}80 & 0\\ 0 & 10\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{8\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 4996057 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{84}_1+A^{1}_1\) ↪ \(C^{1}_5\)
58 out of 119
Subalgebra type: \(\displaystyle A^{84}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{84}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{84}_1\): (14, 24, 30, 32, 16): 168, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 16g_{13}+15g_{3}+12g_{2}+7g_{1}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/42 & 0\\ 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}168 & 0\\ 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{14\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{7\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 626 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2\) ↪ \(C^{1}_5\)
59 out of 119
Subalgebra type: \(\displaystyle B^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) .
Centralizer: \(\displaystyle C^{1}_3\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2\)
Basis of Cartan of centralizer: 3 vectors: (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+A^{1}_1\) , \(\displaystyle B^{1}_2+A^{2}_1\) , \(\displaystyle B^{1}_2+A^{3}_1\) , \(\displaystyle B^{1}_2+A^{8}_1\) , \(\displaystyle B^{1}_2+A^{10}_1\) , \(\displaystyle B^{1}_2+A^{11}_1\) , \(\displaystyle B^{1}_2+A^{35}_1\) , \(\displaystyle B^{1}_2+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{2}_1+A^{1}_1\) , \(\displaystyle B^{1}_2+A^{8}_1+A^{3}_1\) , \(\displaystyle B^{1}_2+A^{10}_1+A^{1}_1\) , \(\displaystyle 2B^{1}_2\) , \(\displaystyle A^{2}_2+B^{1}_2\) , \(\displaystyle B^{1}_2+3A^{1}_1\) , \(\displaystyle 2B^{1}_2+A^{1}_1\) , \(\displaystyle C^{1}_3+B^{1}_2\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 10.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1\\ -1 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2\\ -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus 6V_{\omega_{2}}\oplus 21V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{-2\psi_{2}+4\psi_{3}}\oplus V_{\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{\omega_{2}-\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{1}}
\oplus V_{\omega_{2}+\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{-\psi_{1}+2\psi_{3}}\oplus V_{\psi_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{\psi_{2}}
\oplus V_{\omega_{2}-\psi_{1}+\psi_{2}}\oplus V_{2\psi_{1}-\psi_{2}}\oplus V_{\omega_{2}+\psi_{1}-\psi_{2}}\oplus V_{-\psi_{1}-\psi_{2}+2\psi_{3}}
\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 3V_{0}\oplus V_{\omega_{2}-\psi_{1}}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{\psi_{1}+\psi_{2}-2\psi_{3}}
\oplus V_{\omega_{2}+\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{1}+\psi_{2}}\oplus V_{-\psi_{2}}\oplus V_{-\psi_{1}+2\psi_{2}-2\psi_{3}}
\oplus V_{\psi_{1}-2\psi_{3}}\oplus V_{-2\psi_{1}}\oplus V_{-\psi_{1}+\psi_{2}-2\psi_{3}}\oplus V_{2\psi_{2}-4\psi_{3}}\)
Made total 363 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2\) ↪ \(C^{1}_5\)
60 out of 119
Subalgebra type: \(\displaystyle A^{2}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) .
Centralizer: \(\displaystyle B^{1}_2\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)
Basis of Cartan of centralizer: 3 vectors: (1, 0, -1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_2+A^{1}_1\) , \(\displaystyle A^{2}_2+A^{2}_1\) , \(\displaystyle A^{2}_2+A^{10}_1\) , \(\displaystyle A^{2}_2+2A^{1}_1\) , \(\displaystyle A^{2}_2+B^{1}_2\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2\\ -1/2 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2\\ -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 11V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\psi_{1}+2\psi_{3}}
\oplus V_{\omega_{1}+\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{-2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{2}+\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus V_{\omega_{2}-\psi_{1}-\psi_{2}+2\psi_{3}}\oplus V_{\psi_{1}+2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+\psi_{1}+\psi_{2}-2\psi_{3}}
\oplus 3V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{\omega_{1}-\psi_{2}}\oplus V_{-\psi_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{2}-\psi_{1}+\psi_{2}-2\psi_{3}}
\oplus V_{\omega_{2}-2\psi_{1}-\psi_{2}}\oplus V_{2\psi_{2}-4\psi_{3}}\oplus V_{-\psi_{1}-2\psi_{3}}\oplus V_{-2\psi_{1}-2\psi_{2}}\)
Made total 2599 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{2}_2\) ↪ \(C^{1}_5\)
61 out of 119
Subalgebra type: \(\displaystyle B^{2}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, -1, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle B^{2}_2+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{2}_2\): (2, 4, 4, 4, 2): 4, (-2, -4, -2, 0, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 10.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{10}+g_{2}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}+g_{-10}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2\\ -1/2 & 1/2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -4\\ -4 & 8\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{2}}\oplus 4V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi_{2}}\oplus V_{\omega_{2}+\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{2}+2\psi_{1}}\oplus V_{\omega_{2}-\psi_{1}+2\psi_{2}}
\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}}\oplus 2V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{\omega_{2}+\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}-\psi_{1}-2\psi_{2}}
\oplus V_{-4\psi_{2}}\)
Made total 10926438 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{4}_2\) ↪ \(C^{1}_5\)
62 out of 119
Subalgebra type: \(\displaystyle B^{4}_2\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{4}_1\) .
Centralizer: \(\displaystyle A^{5}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{4}_2\)
Basis of Cartan of centralizer: 1 vectors: (2, 0, -2, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle B^{4}_2+A^{5}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{4}_2\): (2, 4, 6, 8, 4): 8, (0, 0, -4, -8, -4): 16
Dimension of subalgebra generated by predefined or computed generators: 10.
Negative simple generators: \(\displaystyle g_{-18}+g_{-22}\), \(\displaystyle g_{12}+g_{4}\)
Positive simple generators: \(\displaystyle g_{22}+g_{18}\), \(\displaystyle 2g_{-4}+2g_{-12}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & -1/4\\ -1/4 & 1/4\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & -8\\ -8 & 16\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi}\oplus V_{4\psi}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{2\omega_{1}-4\psi}\oplus V_{-4\psi}\)
Made total 6858036 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(3A^{1}_1\) ↪ \(C^{1}_5\)
63 out of 119
Subalgebra type: \(\displaystyle 3A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{1}_1\) .
Centralizer: \(\displaystyle B^{1}_2\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle 4A^{1}_1\) , \(\displaystyle A^{2}_1+3A^{1}_1\) , \(\displaystyle A^{10}_1+3A^{1}_1\) , \(\displaystyle 5A^{1}_1\) , \(\displaystyle B^{1}_2+3A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 2, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 2, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{-23}\), \(\displaystyle g_{-19}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{23}\), \(\displaystyle g_{19}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus V_{2\omega_{1}}\oplus 4V_{\omega_{3}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 10V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{2}}\oplus V_{\omega_{3}-\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}-\psi_{1}+2\psi_{2}}
\oplus V_{\omega_{1}-\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}}\oplus V_{\omega_{3}+\psi_{1}}\oplus V_{\omega_{2}+\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}}
\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus V_{2\omega_{1}}\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{3}-\psi_{1}}\oplus V_{\omega_{2}-\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}}
\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{3}+\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}+\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}-2\psi_{2}}
\oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\)
Made total 456 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1+2A^{1}_1\) ↪ \(C^{1}_5\)
64 out of 119
Subalgebra type: \(\displaystyle A^{2}_1+2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1+A^{1}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, 0, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_1+3A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 4, 4, 2): 4, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}
\oplus 3V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{3}+2\psi_{2}}\oplus V_{\omega_{2}+2\psi_{2}}
\oplus V_{\omega_{1}+\omega_{3}+\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}+2\psi_{2}}
\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{3}-\psi_{1}}
\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}}\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}-2\psi_{2}}\oplus V_{\omega_{3}-2\psi_{2}}
\oplus V_{\omega_{2}-2\psi_{2}}\oplus V_{\omega_{1}-\psi_{1}-2\psi_{2}}\oplus V_{-4\psi_{2}}\)
Made total 1106823 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{2}_1+A^{1}_1\) ↪ \(C^{1}_5\)
65 out of 119
Subalgebra type: \(\displaystyle 2A^{2}_1+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{2}_1\) .
Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, 0, 0, 0), (0, 0, 1, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 4, 4, 2): 4, \(\displaystyle A^{2}_1\): (0, 0, 2, 4, 2): 4, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{-16}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{16}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus 3V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}
\oplus 3V_{2\omega_{1}}\oplus 2V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{2}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{2}+\omega_{3}+\psi_{2}}
\oplus V_{\omega_{1}+\omega_{3}+\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}
\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}-\psi_{1}}\oplus V_{\omega_{2}+\omega_{3}-\psi_{2}}
\oplus 2V_{0}\oplus V_{2\omega_{1}-2\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}}\oplus V_{2\omega_{2}-2\psi_{2}}\)
Made total 1182105 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{3}_1+2A^{1}_1\) ↪ \(C^{1}_5\)
66 out of 119
Subalgebra type: \(\displaystyle A^{3}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{3}_1+A^{1}_1\) .
Centralizer: \(\displaystyle A^{8}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{3}_1\)
Basis of Cartan of centralizer: 1 vectors: (1, 0, 0, 0, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{3}_1\): (2, 4, 6, 6, 3): 6, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-19}+g_{-24}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}+g_{19}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/3 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}6 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus 3V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 3V_{\omega_{1}+\omega_{2}}
\oplus 6V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\psi}\oplus V_{\omega_{1}+\omega_{3}+\psi}\oplus V_{\omega_{1}+\omega_{2}+\psi}\oplus V_{2\omega_{1}+\psi}
\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus 2V_{2\omega_{1}}\oplus V_{\psi}\oplus V_{\omega_{1}+\omega_{3}-\psi}\oplus V_{\omega_{1}+\omega_{2}-\psi}\oplus V_{2\omega_{1}-\psi}
\oplus V_{0}\oplus V_{2\omega_{1}-2\psi}\oplus V_{-\psi}\)
Made total 74480 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{4}_1+A^{1}_1\) ↪ \(C^{1}_5\)
67 out of 119
Subalgebra type: \(\displaystyle 2A^{4}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{4}_1\) .
Centralizer: \(\displaystyle A^{4}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle 2A^{4}_1+A^{1}_1\)
Basis of Cartan of centralizer: 1 vectors: (1, 0, 1, 0, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle 3A^{4}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 6, 8, 4): 8, \(\displaystyle A^{4}_1\): (2, 4, 2, 0, 0): 8, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle -g_{-6}+g_{-7}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{21}+g_{20}\), \(\displaystyle g_{7}-g_{6}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0 & 0\\ 0 & 1/2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0 & 0\\ 0 & 8 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}
\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\omega_{2}+2\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+\psi}\oplus V_{2\omega_{1}+2\omega_{2}}
\oplus V_{2\psi}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-\psi}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi}
\oplus V_{0}\oplus V_{-2\psi}\)
Made total 946 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(3A^{4}_1\) ↪ \(C^{1}_5\)
68 out of 119
Subalgebra type: \(\displaystyle 3A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{4}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle 3A^{4}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 6, 8, 4): 8, \(\displaystyle A^{4}_1\): (2, 4, 2, 0, 0): 8, \(\displaystyle A^{4}_1\): (2, 0, 2, 0, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle -g_{-6}+g_{-7}\), \(\displaystyle -g_{-1}+g_{-3}\)
Positive simple generators: \(\displaystyle g_{21}+g_{20}\), \(\displaystyle g_{7}-g_{6}\), \(\displaystyle g_{3}-g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0 & 0\\ 0 & 1/2 & 0\\ 0 & 0 & 1/2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0 & 0\\ 0 & 8 & 0\\ 0 & 0 & 8\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}
\oplus V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+2\psi}\oplus V_{4\psi}\oplus V_{2\omega_{3}}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-2\psi}\oplus V_{0}\oplus V_{-4\psi}\)
Made total 8530 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{5}_1+2A^{4}_1\) ↪ \(C^{1}_5\)
69 out of 119
Subalgebra type: \(\displaystyle A^{5}_1+2A^{4}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{5}_1+A^{4}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{5}_1\): (2, 4, 6, 8, 5): 10, \(\displaystyle A^{4}_1\): (2, 4, 4, 2, 0): 8, \(\displaystyle A^{4}_1\): (2, 0, 0, 2, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-17}+g_{-18}+g_{-19}\), \(\displaystyle -g_{-10}+g_{-11}\), \(\displaystyle -g_{-1}+g_{-4}\)
Positive simple generators: \(\displaystyle g_{19}+g_{18}+g_{17}\), \(\displaystyle g_{11}-g_{10}\), \(\displaystyle g_{4}-g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/5 & 0 & 0\\ 0 & 1/2 & 0\\ 0 & 0 & 1/2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}10 & 0 & 0\\ 0 & 8 & 0\\ 0 & 0 & 8\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\)
Made total 9833 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+2A^{1}_1\) ↪ \(C^{1}_5\)
70 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1+A^{1}_1\) .
Centralizer: \(\displaystyle A^{3}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{8}_1\)
Basis of Cartan of centralizer: 1 vectors: (2, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-11}+g_{-17}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle 2g_{17}+2g_{11}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{3}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+4\psi}\oplus V_{2\omega_{1}+\omega_{3}+2\psi}\oplus V_{2\omega_{1}+\omega_{2}+2\psi}\oplus V_{4\psi}\oplus V_{4\omega_{1}}
\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{3}-2\psi}
\oplus V_{2\omega_{1}+\omega_{2}-2\psi}\oplus V_{0}\oplus V_{4\omega_{1}-4\psi}\oplus V_{-4\psi}\)
Made total 705 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{3}_1+A^{1}_1\) ↪ \(C^{1}_5\)
71 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1+A^{3}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{3}_1\): (2, 0, 2, 2, 1): 6, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-6}+g_{-21}\), \(\displaystyle g_{-1}+g_{-19}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle 2g_{21}+2g_{6}\), \(\displaystyle g_{19}+g_{1}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0 & 0\\ 0 & 2/3 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0 & 0\\ 0 & 6 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{3}}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi}\oplus V_{4\psi}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}
\oplus V_{\omega_{3}+2\psi}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi}\oplus V_{0}
\oplus V_{\omega_{3}-2\psi}\oplus V_{-4\psi}\)
Made total 640 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{3}_1+A^{2}_1\) ↪ \(C^{1}_5\)
72 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1+A^{2}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1+A^{3}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{3}_1\): (2, 0, 2, 2, 1): 6, \(\displaystyle A^{2}_1\): (0, 0, 0, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-6}+g_{-21}\), \(\displaystyle g_{-1}+g_{-19}\), \(\displaystyle g_{-9}\)
Positive simple generators: \(\displaystyle 2g_{21}+2g_{6}\), \(\displaystyle g_{19}+g_{1}\), \(\displaystyle g_{9}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0 & 0\\ 0 & 2/3 & 0\\ 0 & 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0 & 0\\ 0 & 6 & 0\\ 0 & 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus 2V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus 3V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}
\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}+\psi}\oplus V_{2\omega_{3}+2\psi}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}-\psi}
\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{2\omega_{3}-2\psi}\)
Made total 1169233 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{4}_1+A^{1}_1\) ↪ \(C^{1}_5\)
73 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{4}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1+A^{4}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{4}_1\): (2, 0, 2, 4, 2): 8, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-10}+g_{-18}\), \(\displaystyle g_{-1}+g_{-13}+g_{-19}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 2g_{18}+2g_{10}\), \(\displaystyle g_{19}+g_{13}+g_{1}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0 & 0\\ 0 & 1/2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0 & 0\\ 0 & 8 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{3}}
\oplus V_{\omega_{2}+\omega_{3}}\oplus 2V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\)
Made total 963 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{9}_1+A^{3}_1+A^{1}_1\) ↪ \(C^{1}_5\)
74 out of 119
Subalgebra type: \(\displaystyle A^{9}_1+A^{3}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{9}_1+A^{3}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{9}_1\): (4, 8, 10, 10, 5): 18, \(\displaystyle A^{3}_1\): (2, 0, 0, 2, 1): 6, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-10}+g_{-18}+g_{-19}\), \(\displaystyle g_{-1}+g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{19}+2g_{18}+2g_{10}\), \(\displaystyle g_{13}+g_{1}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/9 & 0 & 0\\ 0 & 2/3 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}18 & 0 & 0\\ 0 & 6 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{3}}
\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\)
Made total 729 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+2A^{1}_1\) ↪ \(C^{1}_5\)
75 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1+A^{1}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{10}_1+3A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{\omega_{3}+2\psi}
\oplus V_{\omega_{2}+2\psi}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}-2\psi}
\oplus V_{0}\oplus V_{\omega_{3}-2\psi}\oplus V_{\omega_{2}-2\psi}\oplus V_{-4\psi}\)
Made total 543 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{2}_1+A^{1}_1\) ↪ \(C^{1}_5\)
76 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+A^{2}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1+A^{2}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 1, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{2}_1\): (0, 0, 2, 4, 2): 4, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-16}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle g_{16}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{3}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}
\oplus 3V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}+\psi}\oplus V_{2\omega_{2}+2\psi}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}+\psi}
\oplus V_{3\omega_{1}+\omega_{2}-\psi}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{2}+\omega_{3}-\psi}
\oplus V_{0}\oplus V_{2\omega_{2}-2\psi}\)
Made total 66639 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+A^{8}_1+A^{3}_1\) ↪ \(C^{1}_5\)
77 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+A^{8}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1+A^{8}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{8}_1\): (0, 0, 4, 8, 4): 16, \(\displaystyle A^{3}_1\): (0, 0, 2, 0, 1): 6
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-8}+g_{-9}\), \(\displaystyle g_{-3}+g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle 2g_{9}+2g_{8}\), \(\displaystyle g_{5}+g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0 & 0\\ 0 & 1/4 & 0\\ 0 & 0 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0 & 0\\ 0 & 16 & 0\\ 0 & 0 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{2}+2\omega_{3}}\oplus V_{3\omega_{1}+2\omega_{2}+\omega_{3}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}
\oplus V_{2\omega_{1}}\)
Made total 1467848 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2A^{10}_1+A^{1}_1\) ↪ \(C^{1}_5\)
78 out of 119
Subalgebra type: \(\displaystyle 2A^{10}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 2A^{10}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{10}_1\): (0, 0, 6, 8, 4): 20, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-3}+g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle 4g_{13}+3g_{3}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0 & 0\\ 0 & 1/5 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0 & 0\\ 0 & 20 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{2}}\oplus V_{3\omega_{1}+3\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{2}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{3}}
\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 624 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{11}_1+2A^{1}_1\) ↪ \(C^{1}_5\)
79 out of 119
Subalgebra type: \(\displaystyle A^{11}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{11}_1+A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{11}_1\): (6, 8, 10, 10, 5): 22, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-1}+g_{-19}+g_{-23}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{23}+g_{19}+3g_{1}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/11 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}22 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}
\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}\)
Made total 786 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{35}_1+2A^{1}_1\) ↪ \(C^{1}_5\)
80 out of 119
Subalgebra type: \(\displaystyle A^{35}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{35}_1+A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{35}_1\): (10, 16, 18, 18, 9): 70, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-19}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 9g_{19}+8g_{2}+5g_{1}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2/35 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}70 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{1}}\oplus V_{5\omega_{1}+\omega_{3}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 624 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{1}_1\) ↪ \(C^{1}_5\)
81 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) .
Centralizer: \(\displaystyle B^{1}_2\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+2A^{1}_1\) , \(\displaystyle B^{1}_2+A^{2}_1+A^{1}_1\) , \(\displaystyle B^{1}_2+A^{10}_1+A^{1}_1\) , \(\displaystyle B^{1}_2+3A^{1}_1\) , \(\displaystyle 2B^{1}_2+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-19}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{19}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 4V_{\omega_{3}}\oplus 4V_{\omega_{2}}\oplus 10V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{2}}\oplus V_{\omega_{3}-\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}-\psi_{1}+2\psi_{2}}
\oplus V_{2\psi_{1}}\oplus V_{\omega_{3}+\psi_{1}}\oplus V_{\omega_{2}+\psi_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}
\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{3}-\psi_{1}}\oplus V_{\omega_{2}-\psi_{1}}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{3}+\psi_{1}-2\psi_{2}}
\oplus V_{\omega_{2}+\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\)
Made total 450 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{2}_1\) ↪ \(C^{1}_5\)
82 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{2}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) .
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 1, 0, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+A^{2}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{2}_1\): (0, 0, 2, 4, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-16}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{16}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 4V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi_{2}}\oplus V_{\omega_{3}+\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{3}+2\psi_{1}}\oplus V_{\omega_{2}+2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}+\psi_{1}}
\oplus V_{\omega_{3}-\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{2}+\omega_{3}-\psi_{1}}\oplus 2V_{0}
\oplus V_{2\omega_{3}-2\psi_{1}}\oplus V_{\omega_{3}+\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}-2\psi_{2}}\oplus V_{\omega_{3}-\psi_{1}-2\psi_{2}}
\oplus V_{-4\psi_{2}}\)
Made total 6168445 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{3}_1\) ↪ \(C^{1}_5\)
83 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) .
Centralizer: \(\displaystyle A^{8}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{3}_1\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1, 0)
Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+A^{8}_1+A^{3}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{3}_1\): (0, 0, 2, 4, 3): 6
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-9}+g_{-19}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{19}+g_{9}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 6V_{2\omega_{3}}\oplus 3V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{3}+2\psi}\oplus V_{2\omega_{3}+\psi}\oplus V_{\omega_{2}+\omega_{3}+\psi}\oplus 2V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{2}}\oplus V_{\psi}\oplus V_{2\omega_{3}-\psi}\oplus V_{\omega_{2}+\omega_{3}-\psi}\oplus V_{0}\oplus V_{2\omega_{3}-2\psi}\oplus V_{-\psi}\)
Made total 7138069 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{8}_1\) ↪ \(C^{1}_5\)
84 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{8}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) .
Centralizer: \(\displaystyle A^{3}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle B^{1}_2+A^{8}_1\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 2, 0, -1)
Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+A^{8}_1+A^{3}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{8}_1\): (0, 0, 4, 8, 4): 16
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-4}+g_{-12}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle 2g_{12}+2g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 1/4\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 16\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{4\omega_{3}}\oplus 2V_{\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\omega_{3}+4\psi}\oplus V_{\omega_{2}+2\omega_{3}+2\psi}\oplus V_{4\psi}\oplus V_{4\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}
\oplus V_{\omega_{2}+2\omega_{3}-2\psi}\oplus V_{0}\oplus V_{4\omega_{3}-4\psi}\oplus V_{-4\psi}\)
Made total 206640 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{10}_1\) ↪ \(C^{1}_5\)
85 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+A^{10}_1+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{10}_1\): (0, 0, 6, 8, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-3}+g_{-13}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle 4g_{13}+3g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{3}}\oplus V_{\omega_{2}+3\omega_{3}}\oplus 2V_{3\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{2}}
\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{6\omega_{3}}\oplus V_{3\omega_{3}+2\psi}\oplus V_{4\psi}\oplus V_{\omega_{2}+3\omega_{3}}\oplus V_{\omega_{2}+2\psi}\oplus V_{2\omega_{3}}
\oplus V_{2\omega_{2}}\oplus V_{3\omega_{3}-2\psi}\oplus V_{0}\oplus V_{\omega_{2}-2\psi}\oplus V_{-4\psi}\)
Made total 2816 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{11}_1\) ↪ \(C^{1}_5\)
86 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{11}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{11}_1\): (0, 0, 6, 8, 5): 22
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-3}+g_{-5}+g_{-13}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle 4g_{13}+g_{5}+3g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2/11\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 22\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{3}}\oplus V_{4\omega_{3}}\oplus V_{\omega_{2}+3\omega_{3}}\oplus 3V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\)
Made total 1149555 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{35}_1\) ↪ \(C^{1}_5\)
87 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{35}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{35}_1\): (0, 0, 10, 16, 9): 70
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-3}+g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle 9g_{5}+8g_{4}+5g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2/35\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 70\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{10\omega_{3}}\oplus V_{6\omega_{3}}\oplus V_{\omega_{2}+5\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\)
Made total 10737 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{1}_1\) ↪ \(C^{1}_5\)
88 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) .
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, -1, -1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_2+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\ -1/2 & 1 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}
\oplus V_{\omega_{2}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+\psi_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}+\omega_{3}-\psi_{1}}
\oplus 2V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{\omega_{1}-2\psi_{2}}\oplus V_{\omega_{3}-\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}}
\oplus V_{-2\psi_{1}-4\psi_{2}}\)
Made total 448 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{2}_1\) ↪ \(C^{1}_5\)
89 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{2}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) .
Centralizer: \(\displaystyle T_{2}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0), (2, 0, -2, 0, -1)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, \(\displaystyle A^{2}_1\): (0, 0, 0, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-9}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{9}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\ -1/2 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus V_{2\omega_{1}}\oplus 2V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{3}+2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+4\psi_{2}}
\oplus V_{\omega_{2}+\omega_{3}+\psi_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{3}-\psi_{1}}
\oplus 2V_{0}\oplus V_{2\omega_{2}-4\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}-\psi_{1}-4\psi_{2}}\oplus V_{2\omega_{3}-2\psi_{1}-4\psi_{2}}\)
Made total 13856855 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+A^{10}_1\) ↪ \(C^{1}_5\)
90 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (2, 0, -2, -2, -1)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, \(\displaystyle A^{10}_1\): (0, 0, 0, 6, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle 4g_{5}+3g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\ -1/2 & 1 & 0\\ 0 & 0 & 1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{3}}\oplus V_{\omega_{2}+3\omega_{3}}\oplus V_{\omega_{1}+3\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi}\oplus V_{\omega_{1}+3\omega_{3}+2\psi}\oplus V_{6\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus V_{\omega_{2}+3\omega_{3}-2\psi}\oplus V_{0}\oplus V_{2\omega_{2}-4\psi}\)
Made total 2814 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{2}_2+A^{1}_1\) ↪ \(C^{1}_5\)
91 out of 119
Subalgebra type: \(\displaystyle B^{2}_2+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{2}_2\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (1, 0, -1, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{2}_2\): (2, 4, 4, 4, 2): 4, (-2, -4, -2, 0, 0): 8, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{10}+g_{2}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}+g_{-10}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\ -1/2 & 1/2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -4 & 0\\ -4 & 8 & 0\\ 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 3V_{2\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{2}+2\psi}\oplus V_{\omega_{2}+\omega_{3}+\psi}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{\omega_{2}+\omega_{3}-\psi}
\oplus V_{0}\oplus V_{2\omega_{2}-2\psi}\)
Made total 93254 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{4}_2+A^{5}_1\) ↪ \(C^{1}_5\)
92 out of 119
Subalgebra type: \(\displaystyle B^{4}_2+A^{5}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{4}_2\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{4}_2\): (2, 4, 6, 8, 4): 8, (0, 0, -4, -8, -4): 16, \(\displaystyle A^{5}_1\): (2, 0, 2, 0, 1): 10
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle g_{9}+g_{8}\), \(\displaystyle g_{-1}-g_{-3}-g_{-5}\)
Positive simple generators: \(\displaystyle g_{21}+g_{20}\), \(\displaystyle 2g_{-8}+2g_{-9}\), \(\displaystyle -g_{5}-g_{3}+g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & -1/4 & 0\\ -1/4 & 1/4 & 0\\ 0 & 0 & 2/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & -8 & 0\\ -8 & 16 & 0\\ 0 & 0 & 10\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\)
Made total 5126406 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3\) ↪ \(C^{1}_5\)
93 out of 119
Subalgebra type: \(\displaystyle C^{1}_3\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) .
Centralizer: \(\displaystyle B^{1}_2\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_3\)
Basis of Cartan of centralizer: 2 vectors: (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle C^{1}_3+A^{1}_1\) , \(\displaystyle C^{1}_3+A^{2}_1\) , \(\displaystyle C^{1}_3+A^{10}_1\) , \(\displaystyle C^{1}_3+2A^{1}_1\) , \(\displaystyle C^{1}_3+B^{1}_2\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, -2, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 21.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{19}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-19}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\ -1/2 & 1 & -1\\ 0 & -1 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\ -2 & 4 & -2\\ 0 & -2 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus 4V_{\omega_{1}}\oplus 10V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{2}}\oplus V_{\omega_{1}-\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}}\oplus V_{\omega_{1}+\psi_{1}}
\oplus V_{2\omega_{1}}\oplus V_{-2\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}-\psi_{1}}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\psi_{1}-2\psi_{2}}
\oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\)
Made total 450 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_3\) ↪ \(C^{1}_5\)
94 out of 119
Subalgebra type: \(\displaystyle A^{2}_3\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2\) .
Centralizer: \(\displaystyle A^{1}_1\) + \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 2 vectors: (1, 0, -1, -2, 0), (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle A^{2}_3+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_3\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, 0, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 15.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{3}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0\\ -1/2 & 1 & -1/2\\ 0 & -1/2 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0\\ -2 & 4 & -2\\ 0 & -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{1}}\oplus 4V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+3\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}
\oplus V_{\omega_{1}+\omega_{3}}\oplus 2V_{0}\oplus V_{2\omega_{3}-2\psi_{1}}\oplus V_{\omega_{1}-\psi_{1}-2\psi_{2}}\oplus V_{\omega_{3}-3\psi_{1}-2\psi_{2}}
\oplus V_{-4\psi_{1}-4\psi_{2}}\)
Made total 446 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(4A^{1}_1\) ↪ \(C^{1}_5\)
95 out of 119
Subalgebra type: \(\displaystyle 4A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle 3A^{1}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle 5A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 2, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 2, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 12.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{-23}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{23}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}
\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}
\oplus 2V_{\omega_{4}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{\omega_{4}+2\psi}\oplus V_{\omega_{3}+2\psi}\oplus V_{\omega_{2}+2\psi}\oplus V_{\omega_{1}+2\psi}\oplus V_{2\omega_{4}}
\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}
\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{4}-2\psi}
\oplus V_{\omega_{3}-2\psi}\oplus V_{\omega_{2}-2\psi}\oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\)
Made total 541 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_1+3A^{1}_1\) ↪ \(C^{1}_5\)
96 out of 119
Subalgebra type: \(\displaystyle A^{2}_1+3A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_1+2A^{1}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (1, 0, 0, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_1\): (2, 4, 4, 4, 2): 4, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 12.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus 2V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}
\oplus V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}
\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\psi}\oplus V_{\omega_{1}+\omega_{4}+\psi}\oplus V_{\omega_{1}+\omega_{3}+\psi}\oplus V_{\omega_{1}+\omega_{2}+\psi}
\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{4}-\psi}\oplus V_{\omega_{1}+\omega_{3}-\psi}\oplus V_{\omega_{1}+\omega_{2}-\psi}
\oplus V_{0}\oplus V_{2\omega_{1}-2\psi}\)
Made total 61833 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(3A^{4}_1+A^{1}_1\) ↪ \(C^{1}_5\)
97 out of 119
Subalgebra type: \(\displaystyle 3A^{4}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle 3A^{4}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{4}_1\): (2, 4, 6, 8, 4): 8, \(\displaystyle A^{4}_1\): (2, 4, 2, 0, 0): 8, \(\displaystyle A^{4}_1\): (2, 0, 2, 0, 0): 8, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 12.
Negative simple generators: \(\displaystyle g_{-20}+g_{-21}\), \(\displaystyle -g_{-6}+g_{-7}\), \(\displaystyle -g_{-1}+g_{-3}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{21}+g_{20}\), \(\displaystyle g_{7}-g_{6}\), \(\displaystyle g_{3}-g_{1}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/2 & 0 & 0 & 0\\ 0 & 1/2 & 0 & 0\\ 0 & 0 & 1/2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}8 & 0 & 0 & 0\\ 0 & 8 & 0 & 0\\ 0 & 0 & 8 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}
\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 802 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{8}_1+A^{3}_1+2A^{1}_1\) ↪ \(C^{1}_5\)
98 out of 119
Subalgebra type: \(\displaystyle A^{8}_1+A^{3}_1+2A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{8}_1+A^{3}_1+A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{8}_1\): (4, 8, 8, 8, 4): 16, \(\displaystyle A^{3}_1\): (2, 0, 2, 2, 1): 6, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 12.
Negative simple generators: \(\displaystyle g_{-6}+g_{-21}\), \(\displaystyle g_{-1}+g_{-19}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 2g_{21}+2g_{6}\), \(\displaystyle g_{19}+g_{1}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/4 & 0 & 0 & 0\\ 0 & 2/3 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}16 & 0 & 0 & 0\\ 0 & 6 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{4}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 723 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{10}_1+3A^{1}_1\) ↪ \(C^{1}_5\)
99 out of 119
Subalgebra type: \(\displaystyle A^{10}_1+3A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle A^{10}_1+2A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{10}_1\): (6, 8, 8, 8, 4): 20, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 12.
Negative simple generators: \(\displaystyle g_{-1}+g_{-23}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle 4g_{23}+3g_{1}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/5 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}20 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{4}}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{4}}
\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}
\oplus V_{2\omega_{1}}\)
Made total 626 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+2A^{1}_1\) ↪ \(C^{1}_5\)
100 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2+A^{1}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle B^{1}_2+3A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 16.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 1 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\ -2 & 4 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{4}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{\omega_{4}+2\psi}\oplus V_{\omega_{3}+2\psi}\oplus V_{\omega_{2}+2\psi}\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}
\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{0}\oplus V_{\omega_{4}-2\psi}
\oplus V_{\omega_{3}-2\psi}\oplus V_{\omega_{2}-2\psi}\oplus V_{-4\psi}\)
Made total 535 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{2}_1+A^{1}_1\) ↪ \(C^{1}_5\)
101 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{2}_1+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2+A^{2}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 1, 0, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{2}_1\): (0, 0, 2, 4, 2): 4, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 16.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-16}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{16}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\ -2 & 4 & 0 & 0\\ 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus 2V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus 3V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}
\oplus V_{2\omega_{2}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{3}+2\psi}\oplus V_{\omega_{3}+\omega_{4}+\psi}\oplus V_{\omega_{2}+\omega_{3}+\psi}\oplus V_{2\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}
\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{3}+\omega_{4}-\psi}\oplus V_{\omega_{2}+\omega_{3}-\psi}\oplus V_{0}\oplus V_{2\omega_{3}-2\psi}\)
Made total 62956 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{8}_1+A^{3}_1\) ↪ \(C^{1}_5\)
102 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{8}_1+A^{3}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2+A^{8}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{8}_1\): (0, 0, 4, 8, 4): 16, \(\displaystyle A^{3}_1\): (0, 0, 2, 0, 1): 6
Dimension of subalgebra generated by predefined or computed generators: 16.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-8}+g_{-9}\), \(\displaystyle g_{-3}+g_{-5}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle 2g_{9}+2g_{8}\), \(\displaystyle g_{5}+g_{3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 1 & 0 & 0\\ 0 & 0 & 1/4 & 0\\ 0 & 0 & 0 & 2/3\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\ -2 & 4 & 0 & 0\\ 0 & 0 & 16 & 0\\ 0 & 0 & 0 & 6\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{4\omega_{3}+2\omega_{4}}\oplus V_{\omega_{2}+2\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\)
Made total 1467538 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+A^{10}_1+A^{1}_1\) ↪ \(C^{1}_5\)
103 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+A^{10}_1+A^{1}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2+A^{10}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{10}_1\): (0, 0, 6, 8, 4): 20, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 16.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-3}+g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle 4g_{13}+3g_{3}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 1 & 0 & 0\\ 0 & 0 & 1/5 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\ -2 & 4 & 0 & 0\\ 0 & 0 & 20 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{3}}\oplus V_{3\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+3\omega_{3}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}
\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\)
Made total 622 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2B^{1}_2\) ↪ \(C^{1}_5\)
104 out of 119
Subalgebra type: \(\displaystyle 2B^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2+A^{1}_1\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle 2B^{1}_2+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle B^{1}_2\): (0, 0, 2, 2, 1): 2, (0, 0, -2, 0, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 20.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{3}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{19}\), \(\displaystyle g_{-3}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\ -1 & 1 & 0 & 0\\ 0 & 0 & 2 & -1\\ 0 & 0 & -1 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\ -2 & 4 & 0 & 0\\ 0 & 0 & 2 & -2\\ 0 & 0 & -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{4}}\oplus 2V_{\omega_{2}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{\omega_{4}+2\psi}\oplus V_{\omega_{2}+2\psi}\oplus V_{2\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{2\omega_{2}}
\oplus V_{0}\oplus V_{\omega_{4}-2\psi}\oplus V_{\omega_{2}-2\psi}\oplus V_{-4\psi}\)
Made total 533 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+2A^{1}_1\) ↪ \(C^{1}_5\)
105 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2+A^{1}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (2, 0, -2, -2, -1)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}
\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}
\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi}\oplus V_{\omega_{1}+\omega_{4}+2\psi}\oplus V_{\omega_{1}+\omega_{3}+2\psi}\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}
\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{4}-2\psi}\oplus V_{\omega_{2}+\omega_{3}-2\psi}
\oplus V_{2\omega_{2}-4\psi}\)
Made total 531 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_2+B^{1}_2\) ↪ \(C^{1}_5\)
106 out of 119
Subalgebra type: \(\displaystyle A^{2}_2+B^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_2+A^{1}_1\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (2, 0, -2, -2, -1)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_2\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, \(\displaystyle B^{1}_2\): (0, 0, 0, 2, 1): 2, (0, 0, 0, -2, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 18.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{4}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{13}\), \(\displaystyle g_{-4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & 0 & 0\\ 0 & 0 & 2 & -1\\ 0 & 0 & -1 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & 0 & 0\\ 0 & 0 & 2 & -2\\ 0 & 0 & -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
\oplus V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi}\oplus V_{\omega_{1}+\omega_{4}+2\psi}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{4}-2\psi}
\oplus V_{2\omega_{2}-4\psi}\)
Made total 539 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3+A^{1}_1\) ↪ \(C^{1}_5\)
107 out of 119
Subalgebra type: \(\displaystyle C^{1}_3+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_3\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle C^{1}_3+2A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, -2, -1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{19}\), \(\displaystyle g_{-13}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & -1 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & -2 & 0\\ 0 & -2 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{4}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{\omega_{4}+2\psi}\oplus V_{\omega_{1}+2\psi}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{1}}
\oplus V_{0}\oplus V_{\omega_{4}-2\psi}\oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\)
Made total 535 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3+A^{2}_1\) ↪ \(C^{1}_5\)
108 out of 119
Subalgebra type: \(\displaystyle C^{1}_3+A^{2}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_3\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1, 0)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, -2, -1): 2, \(\displaystyle A^{2}_1\): (0, 0, 0, 2, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{19}\), \(\displaystyle g_{-9}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{9}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & -1 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & -2 & 0\\ 0 & -2 & 2 & 0\\ 0 & 0 & 0 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle 3V_{2\omega_{4}}\oplus 2V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{4}+2\psi}\oplus V_{\omega_{1}+\omega_{4}+\psi}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}+\omega_{4}-\psi}
\oplus V_{0}\oplus V_{2\omega_{4}-2\psi}\)
Made total 1037773 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3+A^{10}_1\) ↪ \(C^{1}_5\)
109 out of 119
Subalgebra type: \(\displaystyle C^{1}_3+A^{10}_1\) (click on type for detailed printout).
Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_3\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, -2, -1): 2, \(\displaystyle A^{10}_1\): (0, 0, 0, 6, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{19}\), \(\displaystyle g_{-4}+g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-19}\), \(\displaystyle 4g_{5}+3g_{4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & -1 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 1/5\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & -2 & 0\\ 0 & -2 & 2 & 0\\ 0 & 0 & 0 & 20\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{6\omega_{4}}\oplus V_{\omega_{1}+3\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{1}}\)
Made total 3334 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_3+A^{1}_1\) ↪ \(C^{1}_5\)
110 out of 119
Subalgebra type: \(\displaystyle A^{2}_3+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_3\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (1, 0, -1, -2, -1)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_3\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, 0, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 18.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{3}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-3}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & -1/2 & 0\\ 0 & -1/2 & 1 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & -2 & 0\\ 0 & -2 & 4 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}
\oplus V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+2\psi}\oplus V_{\omega_{1}+\omega_{4}+\psi}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{3}+\omega_{4}-\psi}
\oplus V_{0}\oplus V_{2\omega_{3}-2\psi}\)
Made total 529 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_4\) ↪ \(C^{1}_5\)
111 out of 119
Subalgebra type: \(\displaystyle C^{1}_4\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_3\) .
Centralizer: \(\displaystyle A^{1}_1\) .
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_4\)
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \(\displaystyle C^{1}_4+A^{1}_1\) .

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_4\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, 0, 0): 4, (0, 0, 0, -2, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 36.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{3}\), \(\displaystyle g_{13}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-3}\), \(\displaystyle g_{-13}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & -1/2 & 0\\ 0 & -1/2 & 1 & -1\\ 0 & 0 & -1 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & -2 & 0\\ 0 & -2 & 4 & -2\\ 0 & 0 & -2 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{4\psi}\oplus V_{\omega_{1}+2\psi}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}\)
Made total 531 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(A^{2}_4\) ↪ \(C^{1}_5\)
112 out of 119
Subalgebra type: \(\displaystyle A^{2}_4\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_3\) .
Centralizer: \(\displaystyle T_{1}\) (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)
Basis of Cartan of centralizer: 1 vectors: (2, 0, -2, -4, -3)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{2}_4\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, 0, 0): 4, (0, 0, 0, -2, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{3}\), \(\displaystyle g_{4}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-3}\), \(\displaystyle g_{-4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\ -1/2 & 1 & -1/2 & 0\\ 0 & -1/2 & 1 & -1/2\\ 0 & 0 & -1/2 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\ -2 & 4 & -2 & 0\\ 0 & -2 & 4 & -2\\ 0 & 0 & -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{1}}\oplus V_{0}\)
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \(\displaystyle V_{2\omega_{1}+4\psi}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{0}\oplus V_{2\omega_{4}-4\psi}\)
Made total 529 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(5A^{1}_1\) ↪ \(C^{1}_5\)
113 out of 119
Subalgebra type: \(\displaystyle 5A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle 4A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (2, 2, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 2, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 15.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{-23}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{23}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{5}}\oplus V_{\omega_{4}+\omega_{5}}\oplus V_{\omega_{3}+\omega_{5}}\oplus V_{\omega_{2}+\omega_{5}}\oplus V_{\omega_{1}+\omega_{5}}
\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}
\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\)
Made total 624 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(B^{1}_2+3A^{1}_1\) ↪ \(C^{1}_5\)
114 out of 119
Subalgebra type: \(\displaystyle B^{1}_2+3A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle B^{1}_2+2A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 2, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 19.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{19}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\ -1 & 1 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0 & 0\\ -2 & 4 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{5}}\oplus V_{\omega_{4}+\omega_{5}}\oplus V_{\omega_{3}+\omega_{5}}\oplus V_{\omega_{2}+\omega_{5}}\oplus V_{2\omega_{4}}
\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\)
Made total 618 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(2B^{1}_2+A^{1}_1\) ↪ \(C^{1}_5\)
115 out of 119
Subalgebra type: \(\displaystyle 2B^{1}_2+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle 2B^{1}_2\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle B^{1}_2\): (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, \(\displaystyle B^{1}_2\): (0, 0, 2, 2, 1): 2, (0, 0, -2, 0, 0): 4, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 23.
Negative simple generators: \(\displaystyle g_{-25}\), \(\displaystyle g_{1}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{3}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{25}\), \(\displaystyle g_{-1}\), \(\displaystyle g_{19}\), \(\displaystyle g_{-3}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\ -1 & 1 & 0 & 0 & 0\\ 0 & 0 & 2 & -1 & 0\\ 0 & 0 & -1 & 1 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -2 & 0 & 0 & 0\\ -2 & 4 & 0 & 0 & 0\\ 0 & 0 & 2 & -2 & 0\\ 0 & 0 & -2 & 4 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{5}}\oplus V_{\omega_{4}+\omega_{5}}\oplus V_{\omega_{2}+\omega_{5}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}}
\oplus V_{2\omega_{2}}\)
Made total 616 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3+2A^{1}_1\) ↪ \(C^{1}_5\)
116 out of 119
Subalgebra type: \(\displaystyle C^{1}_3+2A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_3+A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, -2, -1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 2, 1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 27.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{19}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0 & 0\\ -1/2 & 1 & -1 & 0 & 0\\ 0 & -1 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0 & 0\\ -2 & 4 & -2 & 0 & 0\\ 0 & -2 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{5}}\oplus V_{\omega_{4}+\omega_{5}}\oplus V_{\omega_{1}+\omega_{5}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}
\oplus V_{2\omega_{1}}\)
Made total 618 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_3+B^{1}_2\) ↪ \(C^{1}_5\)
117 out of 119
Subalgebra type: \(\displaystyle C^{1}_3+B^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_3+A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_3\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, -2, -1): 2, \(\displaystyle B^{1}_2\): (0, 0, 0, 2, 1): 2, (0, 0, 0, -2, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 31.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{19}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{4}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-19}\), \(\displaystyle g_{13}\), \(\displaystyle g_{-4}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0 & 0\\ -1/2 & 1 & -1 & 0 & 0\\ 0 & -1 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & -1\\ 0 & 0 & 0 & -1 & 1\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0 & 0\\ -2 & 4 & -2 & 0 & 0\\ 0 & -2 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & -2\\ 0 & 0 & 0 & -2 & 4\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{5}}\oplus V_{\omega_{1}+\omega_{5}}\oplus V_{2\omega_{1}}\)
Made total 618 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_4+A^{1}_1\) ↪ \(C^{1}_5\)
118 out of 119
Subalgebra type: \(\displaystyle C^{1}_4+A^{1}_1\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle C^{1}_4\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_4\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, 0, 0): 4, (0, 0, 0, -2, -1): 2, \(\displaystyle A^{1}_1\): (0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 39.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{3}\), \(\displaystyle g_{13}\), \(\displaystyle g_{-5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-3}\), \(\displaystyle g_{-13}\), \(\displaystyle g_{5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0\\ 0 & -1/2 & 1 & -1 & 0\\ 0 & 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0 & 0\\ -2 & 4 & -2 & 0 & 0\\ 0 & -2 & 4 & -2 & 0\\ 0 & 0 & -2 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{5}}\oplus V_{\omega_{1}+\omega_{5}}\oplus V_{2\omega_{1}}\)
Made total 614 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra \(C^{1}_5\) ↪ \(C^{1}_5\)
119 out of 119
Subalgebra type: \(\displaystyle C^{1}_5\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{2}_4\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle C^{1}_5\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle C^{1}_5\): (2, 4, 4, 4, 2): 4, (0, -2, 0, 0, 0): 4, (0, 0, -2, 0, 0): 4, (0, 0, 0, -2, 0): 4, (0, 0, 0, 0, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 55.
Negative simple generators: \(\displaystyle g_{-24}\), \(\displaystyle g_{2}\), \(\displaystyle g_{3}\), \(\displaystyle g_{4}\), \(\displaystyle g_{5}\)
Positive simple generators: \(\displaystyle g_{24}\), \(\displaystyle g_{-2}\), \(\displaystyle g_{-3}\), \(\displaystyle g_{-4}\), \(\displaystyle g_{-5}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0\\ 0 & -1/2 & 1 & -1/2 & 0\\ 0 & 0 & -1/2 & 1 & -1\\ 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}4 & -2 & 0 & 0 & 0\\ -2 & 4 & -2 & 0 & 0\\ 0 & -2 & 4 & -2 & 0\\ 0 & 0 & -2 & 4 & -2\\ 0 & 0 & 0 & -2 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\)
Made total 612 arithmetic operations while solving the Serre relations polynomial system.
Of the 23 h element conjugacy classes 14 had their Weyl group orbits computed and buffered. The h elements and their computed orbit sizes follow.
h elementorbit size
(18, 32, 42, 48, 25)size not computed
(14, 24, 30, 32, 17)size not computed
(14, 24, 30, 32, 16)size not computed
(10, 16, 22, 24, 13)960
(8, 16, 20, 24, 12)size not computed
(10, 16, 18, 20, 11)size not computed
(10, 16, 18, 20, 10)size not computed
(10, 16, 18, 18, 9)480
(6, 12, 14, 16, 9)size not computed
(6, 12, 14, 16, 8)480
(6, 10, 14, 16, 8)960
(6, 8, 10, 12, 7)size not computed
(6, 8, 10, 12, 6)size not computed
(6, 8, 10, 10, 5)240
(6, 8, 8, 8, 4)80
(4, 8, 10, 12, 6)480
(4, 8, 10, 10, 5)240
(4, 8, 8, 8, 4)40
(2, 4, 6, 8, 5)32
(2, 4, 6, 8, 4)80
(2, 4, 6, 6, 3)80
(2, 4, 4, 4, 2)40
(2, 2, 2, 2, 1)10

Number of sl(2) subalgebras: 23.
Let h be in the Cartan subalgebra. Let \(\alpha_1, ..., \alpha_n\) be simple roots with respect to h. Then the h-characteristic, as defined by E. Dynkin, is the n-tuple \((\alpha_1(h), ..., \alpha_n(h))\).

The actual realization of h. The coordinates of h are given with respect to the fixed original simple basis. Note that the h-characteristic is computed using a possibly different simple basis, more precisely, with respect to any h-positive simple basis.
A regular semisimple subalgebra might contain an sl(2) such that the sl(2) has no centralizer in the regular semisimple subalgebra, but the regular semisimple subalgebra might fail to be minimal containing. This happens when another minimal containing regular semisimple subalgebra of equal rank nests as a root subalgebra in the containing SA. See Dynkin, Semisimple Lie subalgebras of semisimple Lie algebras, remark before Theorem 10.4.
The \(sl(2)\) submodules of the ambient Lie algebra are parametrized by their highest weight with respect to the Cartan element h of \(sl(2)\). In turn, the highest weight is a positive integer multiple of the fundamental highest weight \(\psi\). \(V_{l\psi}\) is \(l + 1\)-dimensional.


Type + realization linkh-CharacteristicRealization of hsl(2)-module decomposition of the ambient Lie algebra
\(\psi=\) the fundamental \(sl(2)\)-weight. 		Centralizer dimensionType of semisimple part of centralizer, if knownThe square of the length of the weight dual to h.Dynkin index Minimal containing regular semisimple SAsContaining regular semisimple SAs in which the sl(2) has no centralizer
\(A^{165}_1\)(2, 2, 2, 2, 2)(18, 32, 42, 48, 25)\(V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}\)
0 \(\displaystyle 0\)330165C^{1}_5; C^{1}_5;
\(A^{85}_1\)(2, 2, 2, 0, 2)(14, 24, 30, 32, 17)\(V_{14\psi}+V_{10\psi}+V_{8\psi}+2V_{6\psi}+2V_{2\psi}\)
0 \(\displaystyle 0\)17085C^{1}_5; C^{1}_4+A^{1}_1; C^{1}_5; C^{1}_4+A^{1}_1;
\(A^{84}_1\)(2, 2, 2, 1, 0)(14, 24, 30, 32, 16)\(V_{14\psi}+V_{10\psi}+2V_{7\psi}+V_{6\psi}+V_{2\psi}+3V_{0}\)
3 \(\displaystyle A^{1}_1\)16884C^{1}_4; C^{1}_4;
\(A^{45}_1\)(2, 0, 2, 0, 2)(10, 16, 22, 24, 13)\(V_{10\psi}+V_{8\psi}+3V_{6\psi}+V_{4\psi}+3V_{2\psi}\)
0 \(\displaystyle 0\)9045C^{1}_5; C^{1}_3+B^{1}_2; C^{1}_5; C^{1}_3+B^{1}_2;
\(A^{40}_1\)(0, 2, 0, 2, 0)(8, 16, 20, 24, 12)\(3V_{8\psi}+V_{6\psi}+3V_{4\psi}+V_{2\psi}+3V_{0}\)
3 not computed8040A^{2}_4; A^{2}_4;
\(A^{37}_1\)(2, 2, 0, 0, 2)(10, 16, 18, 20, 11)\(V_{10\psi}+3V_{6\psi}+2V_{4\psi}+4V_{2\psi}+V_{0}\)
1 \(\displaystyle 0\)7437C^{1}_4+A^{1}_1; C^{1}_3+2A^{1}_1; C^{1}_3+A^{2}_1; C^{1}_4+A^{1}_1; C^{1}_3+2A^{1}_1; C^{1}_3+A^{2}_1;
\(A^{36}_1\)(2, 2, 0, 1, 0)(10, 16, 18, 20, 10)\(V_{10\psi}+2V_{6\psi}+2V_{5\psi}+V_{4\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}\)
3 \(\displaystyle A^{1}_1\)7236C^{1}_4; C^{1}_3+A^{1}_1; C^{1}_4; C^{1}_3+A^{1}_1;
\(A^{35}_1\)(2, 2, 1, 0, 0)(10, 16, 18, 18, 9)\(V_{10\psi}+V_{6\psi}+4V_{5\psi}+V_{2\psi}+10V_{0}\)
10 \(\displaystyle B^{1}_2\)7035C^{1}_3; C^{1}_3;
\(A^{21}_1\)(0, 2, 0, 0, 2)(6, 12, 14, 16, 9)\(3V_{6\psi}+3V_{4\psi}+6V_{2\psi}+V_{0}\)
1 \(\displaystyle 0\)4221C^{1}_3+B^{1}_2; 2B^{1}_2+A^{1}_1; A^{2}_3+A^{1}_1; C^{1}_3+B^{1}_2; 2B^{1}_2+A^{1}_1; A^{2}_3+A^{1}_1;
\(A^{20}_1\)(0, 2, 0, 1, 0)(6, 12, 14, 16, 8)\(3V_{6\psi}+V_{4\psi}+4V_{3\psi}+3V_{2\psi}+4V_{0}\)
4 \(\displaystyle A^{1}_1\)40202B^{1}_2; A^{2}_3; 2B^{1}_2; A^{2}_3;
\(A^{18}_1\)(1, 0, 1, 1, 0)(6, 10, 14, 16, 8)\(V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}\)
3 not computed3618A^{2}_2+B^{1}_2; A^{2}_2+B^{1}_2;
\(A^{13}_1\)(2, 0, 0, 0, 2)(6, 8, 10, 12, 7)\(V_{6\psi}+3V_{4\psi}+10V_{2\psi}+3V_{0}\)
3 not computed2613C^{1}_3+2A^{1}_1; B^{1}_2+3A^{1}_1; C^{1}_3+A^{2}_1; B^{1}_2+A^{2}_1+A^{1}_1; C^{1}_3+2A^{1}_1; B^{1}_2+3A^{1}_1; C^{1}_3+A^{2}_1; B^{1}_2+A^{2}_1+A^{1}_1;
\(A^{12}_1\)(2, 0, 0, 1, 0)(6, 8, 10, 12, 6)\(V_{6\psi}+2V_{4\psi}+2V_{3\psi}+6V_{2\psi}+4V_{\psi}+4V_{0}\)
4 \(\displaystyle A^{1}_1\)2412C^{1}_3+A^{1}_1; B^{1}_2+2A^{1}_1; B^{1}_2+A^{2}_1; C^{1}_3+A^{1}_1; B^{1}_2+2A^{1}_1; B^{1}_2+A^{2}_1;
\(A^{11}_1\)(2, 0, 1, 0, 0)(6, 8, 10, 10, 5)\(V_{6\psi}+V_{4\psi}+4V_{3\psi}+3V_{2\psi}+4V_{\psi}+10V_{0}\)
10 \(\displaystyle B^{1}_2\)2211C^{1}_3; B^{1}_2+A^{1}_1; C^{1}_3; B^{1}_2+A^{1}_1;
\(A^{10}_1\)(2, 1, 0, 0, 0)(6, 8, 8, 8, 4)\(V_{6\psi}+6V_{3\psi}+V_{2\psi}+21V_{0}\)
21 \(\displaystyle C^{1}_3\)2010B^{1}_2; B^{1}_2;
\(A^{10}_1\)(0, 1, 0, 1, 0)(4, 8, 10, 12, 6)\(3V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi}+4V_{0}\)
4 not computed2010A^{2}_2+2A^{1}_1; A^{2}_2+A^{2}_1; A^{2}_2+2A^{1}_1; A^{2}_2+A^{2}_1;
\(A^{9}_1\)(0, 1, 1, 0, 0)(4, 8, 10, 10, 5)\(3V_{4\psi}+2V_{3\psi}+6V_{2\psi}+4V_{\psi}+6V_{0}\)
6 not computed189A^{2}_2+A^{1}_1; A^{2}_2+A^{1}_1;
\(A^{8}_1\)(0, 2, 0, 0, 0)(4, 8, 8, 8, 4)\(3V_{4\psi}+9V_{2\psi}+13V_{0}\)
13 not computed168A^{2}_2; A^{2}_2;
\(A^{5}_1\)(0, 0, 0, 0, 2)(2, 4, 6, 8, 5)\(15V_{2\psi}+10V_{0}\)
10 not computed1055A^{1}_1; A^{2}_1+3A^{1}_1; 2A^{2}_1+A^{1}_1; 5A^{1}_1; A^{2}_1+3A^{1}_1; 2A^{2}_1+A^{1}_1;
\(A^{4}_1\)(0, 0, 0, 1, 0)(2, 4, 6, 8, 4)\(10V_{2\psi}+8V_{\psi}+9V_{0}\)
9 not computed844A^{1}_1; A^{2}_1+2A^{1}_1; 2A^{2}_1; 4A^{1}_1; A^{2}_1+2A^{1}_1; 2A^{2}_1;
\(A^{3}_1\)(0, 0, 1, 0, 0)(2, 4, 6, 6, 3)\(6V_{2\psi}+12V_{\psi}+13V_{0}\)
13 not computed633A^{1}_1; A^{2}_1+A^{1}_1; 3A^{1}_1; A^{2}_1+A^{1}_1;
\(A^{2}_1\)(0, 1, 0, 0, 0)(2, 4, 4, 4, 2)\(3V_{2\psi}+12V_{\psi}+22V_{0}\)
22 \(\displaystyle C^{1}_3\)422A^{1}_1; A^{2}_1; 2A^{1}_1; A^{2}_1;
\(A^{1}_1\)(1, 0, 0, 0, 0)(2, 2, 2, 2, 1)\(V_{2\psi}+8V_{\psi}+36V_{0}\)
36 \(\displaystyle C^{1}_4\)21A^{1}_1; A^{1}_1;

Length longest root ambient algebra squared/4= 1/2

Given a root subsystem P, and a root subsubsystem P_0, in (10.2) of Semisimple subalgebras of semisimple Lie algebras, E. Dynkin defines a numerical constant e(P, P_0) (which we call Dynkin epsilon).
In Theorem 10.3, Dynkin proves that if an sl(2) is an S-subalgebra in the root subalgebra generated by P, such that it has characteristic 2 for all simple roots of P lying in P_0, then e(P, P_0)= 0. It turns out by direct computation that, in the current case of C^{1}_5, e(P,P_0)= 0 implies that an S-sl(2) subalgebra of the root subalgebra generated by P with characteristic with 2's in the simple roots of P_0 always exists. Note that Theorem 10.3 is stated in one direction only.

h-characteristic: (2, 2, 2, 2, 2)
Length of the weight dual to h: 330
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_5
sl(2)-module decomposition of the ambient Lie algebra: \(V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 25h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\)
\( e = 25/17g_{5}+12/5g_{4}+21/5g_{3}+8g_{2}+9g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{6} -18~\\2x_{2} x_{7} -32~\\2x_{3} x_{8} -42~\\2x_{4} x_{9} -48~\\x_{5} x_{10} -25~\\\end{array}\)

h-characteristic: (2, 2, 2, 0, 2)
Length of the weight dual to h: 170
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_5 Containing regular semisimple subalgebra number 2: C^{1}_4+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{10\psi}+V_{8\psi}+2V_{6\psi}+2V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 17h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 841/2074g_{13}+746/3111g_{9}+209/183g_{8}+40/3111g_{5}+100/183g_{3}+6g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{4} x_{13} -x_{3} x_{12} ~\\x_{7} x_{13} +x_{6} x_{11} -x_{5} x_{10} ~\\2x_{1} x_{8} -14~\\2x_{2} x_{9} -24~\\2x_{5} x_{12} +2x_{3} x_{10} -30~\\2x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} -32~\\x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} -17~\\\end{array}\)

h-characteristic: (2, 2, 2, 1, 0)
Length of the weight dual to h: 168
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_4
sl(2)-module decomposition of the ambient Lie algebra: \(V_{14\psi}+V_{10\psi}+2V_{7\psi}+V_{6\psi}+V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 16h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\)
\( e = 8/5g_{13}+3g_{3}+6g_{2}+7g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -14~\\2x_{2} x_{6} -24~\\2x_{3} x_{7} -30~\\2x_{4} x_{8} -32~\\x_{4} x_{8} -16~\\\end{array}\)

h-characteristic: (2, 0, 2, 0, 2)
Length of the weight dual to h: 90
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_5 Containing regular semisimple subalgebra number 2: C^{1}_3+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{8\psi}+3V_{6\psi}+V_{4\psi}+3V_{2\psi}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 13h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\)
\( e = 738295/8225937g_{13}+30275/598823g_{11}+1498759/8225937g_{9}+497596/1796469g_{8}+301843/1197646g_{7}+42425/31517g_{6} \\ -117884/8225937g_{5}-13030/1796469g_{3}+11516/31517g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{17} +x_{2} x_{14} -x_{1} x_{13} ~\\x_{7} x_{18} -x_{5} x_{17} +x_{3} x_{16} -x_{2} x_{15} ~\\x_{8} x_{15} +x_{5} x_{11} -x_{4} x_{10} ~\\x_{9} x_{16} -x_{8} x_{14} +x_{7} x_{12} -x_{6} x_{11} ~\\2x_{4} x_{13} +2x_{1} x_{10} -10~\\2x_{6} x_{15} +2x_{2} x_{11} +2x_{1} x_{10} -16~\\2x_{8} x_{17} +2x_{6} x_{15} +2x_{5} x_{14} +2x_{2} x_{11} -22~\\2x_{7} x_{16} +2x_{5} x_{14} +2x_{3} x_{12} +2x_{2} x_{11} -24~\\x_{9} x_{18} +2x_{7} x_{16} +x_{3} x_{12} -13~\\\end{array}\)

h-characteristic: (0, 2, 0, 2, 0)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_4
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{8\psi}+V_{6\psi}+3V_{4\psi}+V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 12h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\)
\( e = 4g_{10}+3g_{9}+6/5g_{8}+2/5g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -8~\\2x_{4} x_{8} +2x_{1} x_{5} -16~\\2x_{3} x_{7} +2x_{1} x_{5} -20~\\2x_{3} x_{7} +2x_{2} x_{6} -24~\\2x_{2} x_{6} -12~\\\end{array}\)

h-characteristic: (2, 2, 0, 0, 2)
Length of the weight dual to h: 74
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_4+A^{1}_1 Containing regular semisimple subalgebra number 2: C^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_3+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+3V_{6\psi}+2V_{4\psi}+4V_{2\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 11h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\)
\( e = 1/10g_{19}+1010835/1650251g_{13}+4128/3361g_{11}+220938/1650251g_{9}+1096/3361g_{7}-1093/1650251g_{5} \\ +5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{3} x_{13} -x_{2} x_{12} ~\\x_{7} x_{13} +x_{6} x_{10} -x_{5} x_{9} ~\\2x_{1} x_{8} -10~\\2x_{5} x_{12} +2x_{2} x_{9} -16~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{2} x_{9} -18~\\2x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} -20~\\x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} -11~\\\end{array}\)

h-characteristic: (2, 2, 0, 1, 0)
Length of the weight dual to h: 72
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_4 Containing regular semisimple subalgebra number 2: C^{1}_3+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+2V_{6\psi}+2V_{5\psi}+V_{4\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 10h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\)
\( e = 2606/5671g_{19}+1313/5671g_{16}-37/5671g_{13}+148/107g_{7}+56/107g_{2}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{12} +x_{3} x_{11} -x_{2} x_{10} ~\\x_{6} x_{11} +x_{5} x_{9} -x_{4} x_{8} ~\\2x_{1} x_{7} -10~\\2x_{4} x_{10} +2x_{2} x_{8} -16~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -18~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} -20~\\x_{6} x_{12} +2x_{5} x_{11} +x_{3} x_{9} -10~\\\end{array}\)

h-characteristic: (2, 2, 1, 0, 0)
Length of the weight dual to h: 70
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: C^{1}_3
sl(2)-module decomposition of the ambient Lie algebra: \(V_{10\psi}+V_{6\psi}+4V_{5\psi}+V_{2\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 9h_{5}+18h_{4}+18h_{3}+16h_{2}+10h_{1}\)
\( e = 9/5g_{19}+4g_{2}+5g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{4} -10~\\2x_{2} x_{5} -16~\\2x_{3} x_{6} -18~\\2x_{3} x_{6} -18~\\x_{3} x_{6} -9~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 2)
Length of the weight dual to h: 42
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_3+B^{1}_2 Containing regular semisimple subalgebra number 2: 2B^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 3: A^{2}_3+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+3V_{4\psi}+6V_{2\psi}+V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 9h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\)
\( e = 4/5g_{19}+1437/1499g_{14}+466471/902398g_{13}+180/1499g_{10}+34838/451199g_{9}-596/451199g_{5}+3/10g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{2} x_{13} -x_{1} x_{12} ~\\x_{7} x_{13} +x_{6} x_{9} -x_{5} x_{8} ~\\2x_{5} x_{12} +2x_{1} x_{8} -6~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{1} x_{8} -12~\\2x_{5} x_{12} +2x_{3} x_{10} +2x_{1} x_{8} -14~\\2x_{6} x_{13} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -16~\\x_{7} x_{14} +2x_{6} x_{13} +x_{3} x_{10} +x_{2} x_{9} -9~\\\end{array}\)

h-characteristic: (0, 2, 0, 1, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2B^{1}_2 Containing regular semisimple subalgebra number 2: A^{2}_3
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{6\psi}+V_{4\psi}+4V_{3\psi}+3V_{2\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\)
\( e = 4g_{19}+4/5g_{13}+3/10g_{7}+3/2g_{6}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{6} -6~\\2x_{4} x_{8} +2x_{2} x_{6} -12~\\2x_{4} x_{8} +2x_{1} x_{5} -14~\\2x_{3} x_{7} +2x_{1} x_{5} -16~\\x_{3} x_{7} +x_{1} x_{5} -8~\\\end{array}\)

h-characteristic: (1, 0, 1, 1, 0)
Length of the weight dual to h: 36
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_2+B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 8h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\)
\( e = 2g_{15}+4/5g_{13}+3/10g_{10}+g_{8}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{4} x_{8} -6~\\2x_{4} x_{8} +2x_{1} x_{5} -10~\\2x_{4} x_{8} +2x_{2} x_{6} +2x_{1} x_{5} -14~\\2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -16~\\x_{3} x_{7} +2x_{1} x_{5} -8~\\\end{array}\)

h-characteristic: (2, 0, 0, 0, 2)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: C^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 2: B^{1}_2+3A^{1}_1 Containing regular semisimple subalgebra number 3: C^{1}_3+A^{2}_1 Containing regular semisimple subalgebra number 4: B^{1}_2+A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+3V_{4\psi}+10V_{2\psi}+3V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 7h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\)
\( e = 1/5g_{23}+1/10g_{19}+1437/1499g_{14}+466471/902398g_{13}+180/1499g_{10}+34838/451199g_{9}-596/451199g_{5}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{6} x_{14} +x_{2} x_{13} -x_{1} x_{12} ~\\x_{7} x_{13} +x_{6} x_{9} -x_{5} x_{8} ~\\2x_{5} x_{12} +2x_{1} x_{8} -6~\\2x_{5} x_{12} +2x_{3} x_{10} +2x_{1} x_{8} -8~\\2x_{5} x_{12} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{1} x_{8} -10~\\2x_{6} x_{13} +2x_{4} x_{11} +2x_{3} x_{10} +2x_{2} x_{9} +2x_{1} x_{8} -12~\\x_{7} x_{14} +2x_{6} x_{13} +x_{4} x_{11} +x_{3} x_{10} +x_{2} x_{9} -7~\\\end{array}\)

h-characteristic: (2, 0, 0, 1, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: C^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 2: B^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 3: B^{1}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+2V_{4\psi}+2V_{3\psi}+6V_{2\psi}+4V_{\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\)
\( e = 1/5g_{23}+28192/67071g_{19}+8366/67071g_{16}-421/134142g_{13}+294/283g_{10}+111/566g_{6}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{5} x_{12} +x_{2} x_{11} -x_{1} x_{10} ~\\x_{6} x_{11} +x_{5} x_{8} -x_{4} x_{7} ~\\2x_{4} x_{10} +2x_{1} x_{7} -6~\\2x_{4} x_{10} +2x_{3} x_{9} +2x_{1} x_{7} -8~\\2x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} +2x_{1} x_{7} -10~\\2x_{6} x_{12} +4x_{5} x_{11} +2x_{3} x_{9} +2x_{2} x_{8} -12~\\x_{6} x_{12} +2x_{5} x_{11} +x_{3} x_{9} +x_{2} x_{8} -6~\\\end{array}\)

h-characteristic: (2, 0, 1, 0, 0)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: C^{1}_3 Containing regular semisimple subalgebra number 2: B^{1}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+V_{4\psi}+4V_{3\psi}+3V_{2\psi}+4V_{\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 5h_{5}+10h_{4}+10h_{3}+8h_{2}+6h_{1}\)
\( e = 2111/11022g_{23}+1330/5511g_{21}-68/5511g_{19}+201/167g_{6}+60/167g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}x_{4} x_{10} +x_{2} x_{9} -x_{1} x_{8} ~\\x_{5} x_{9} +x_{4} x_{7} -x_{3} x_{6} ~\\2x_{3} x_{8} +2x_{1} x_{6} -6~\\2x_{4} x_{9} +2x_{2} x_{7} +2x_{1} x_{6} -8~\\2x_{5} x_{10} +4x_{4} x_{9} +2x_{2} x_{7} -10~\\2x_{5} x_{10} +4x_{4} x_{9} +2x_{2} x_{7} -10~\\x_{5} x_{10} +2x_{4} x_{9} +x_{2} x_{7} -5~\\\end{array}\)

h-characteristic: (2, 1, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: B^{1}_2
sl(2)-module decomposition of the ambient Lie algebra: \(V_{6\psi}+6V_{3\psi}+V_{2\psi}+21V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{5}+8h_{4}+8h_{3}+8h_{2}+6h_{1}\)
\( e = 4g_{23}+3/2g_{1}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{2} x_{4} -6~\\2x_{1} x_{3} -8~\\2x_{1} x_{3} -8~\\2x_{1} x_{3} -8~\\x_{1} x_{3} -4~\\\end{array}\)

h-characteristic: (0, 1, 0, 1, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A^{2}_2+2A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_2+A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi}+4V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 6h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\)
\( e = 1/5g_{19}+2g_{17}+1/10g_{13}+g_{11}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -4~\\2x_{2} x_{6} +2x_{1} x_{5} -8~\\2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -10~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -12~\\x_{4} x_{8} +x_{3} x_{7} +2x_{1} x_{5} -6~\\\end{array}\)

h-characteristic: (0, 1, 1, 0, 0)
Length of the weight dual to h: 18
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_2+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+2V_{3\psi}+6V_{2\psi}+4V_{\psi}+6V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 5h_{5}+10h_{4}+10h_{3}+8h_{2}+4h_{1}\)
\( e = 2g_{20}+1/5g_{19}+g_{7}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{4} -4~\\2x_{2} x_{5} +2x_{1} x_{4} -8~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -10~\\2x_{3} x_{6} +4x_{1} x_{4} -10~\\x_{3} x_{6} +2x_{1} x_{4} -5~\\\end{array}\)

h-characteristic: (0, 2, 0, 0, 0)
Length of the weight dual to h: 16
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{2}_2
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{4\psi}+9V_{2\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{5}+8h_{4}+8h_{3}+8h_{2}+4h_{1}\)
\( e = 2g_{22}+g_{2}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{3} -4~\\2x_{2} x_{4} +2x_{1} x_{3} -8~\\4x_{1} x_{3} -8~\\4x_{1} x_{3} -8~\\2x_{1} x_{3} -4~\\\end{array}\)

h-characteristic: (0, 0, 0, 0, 2)
Length of the weight dual to h: 10
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 5A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1+3A^{1}_1 Containing regular semisimple subalgebra number 3: 2A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(15V_{2\psi}+10V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 5h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\)
\( e = g_{25}+1/2g_{23}+1/5g_{19}+1/10g_{13}+1/17g_{5}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{6} -2~\\2x_{2} x_{7} +2x_{1} x_{6} -4~\\2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -6~\\2x_{4} x_{9} +2x_{3} x_{8} +2x_{2} x_{7} +2x_{1} x_{6} -8~\\x_{5} x_{10} +x_{4} x_{9} +x_{3} x_{8} +x_{2} x_{7} +x_{1} x_{6} -5~\\\end{array}\)

h-characteristic: (0, 0, 0, 1, 0)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 4A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1+2A^{1}_1 Containing regular semisimple subalgebra number 3: 2A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(10V_{2\psi}+8V_{\psi}+9V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 4h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\)
\( e = g_{25}+1/2g_{23}+1/5g_{19}+1/10g_{13}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{5} -2~\\2x_{2} x_{6} +2x_{1} x_{5} -4~\\2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -6~\\2x_{4} x_{8} +2x_{3} x_{7} +2x_{2} x_{6} +2x_{1} x_{5} -8~\\x_{4} x_{8} +x_{3} x_{7} +x_{2} x_{6} +x_{1} x_{5} -4~\\\end{array}\)

h-characteristic: (0, 0, 1, 0, 0)
Length of the weight dual to h: 6
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 3A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1+A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(6V_{2\psi}+12V_{\psi}+13V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 3h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\)
\( e = g_{25}+1/2g_{23}+1/5g_{19}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{4} -2~\\2x_{2} x_{5} +2x_{1} x_{4} -4~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -6~\\2x_{3} x_{6} +2x_{2} x_{5} +2x_{1} x_{4} -6~\\x_{3} x_{6} +x_{2} x_{5} +x_{1} x_{4} -3~\\\end{array}\)

h-characteristic: (0, 1, 0, 0, 0)
Length of the weight dual to h: 4
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2A^{1}_1 Containing regular semisimple subalgebra number 2: A^{2}_1
sl(2)-module decomposition of the ambient Lie algebra: \(3V_{2\psi}+12V_{\psi}+22V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = 2h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\)
\( e = g_{25}+1/2g_{23}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{3} -2~\\2x_{2} x_{4} +2x_{1} x_{3} -4~\\2x_{2} x_{4} +2x_{1} x_{3} -4~\\2x_{2} x_{4} +2x_{1} x_{3} -4~\\x_{2} x_{4} +x_{1} x_{3} -2~\\\end{array}\)

h-characteristic: (1, 0, 0, 0, 0)
Length of the weight dual to h: 2
Simple basis ambient algebra w.r.t defining h: 5 vectors: (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A^{1}_1
sl(2)-module decomposition of the ambient Lie algebra: \(V_{2\psi}+8V_{\psi}+36V_{0}\)
Below is one possible realization of the sl(2) subalgebra.
\( h = h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\)
\( e = g_{25}
\)
The polynomial system that corresponds to finding the h, e, f triple:
\(\begin{array}{l}2x_{1} x_{2} -2~\\2x_{1} x_{2} -2~\\2x_{1} x_{2} -2~\\2x_{1} x_{2} -2~\\x_{1} x_{2} -1~\\\end{array}\)
Calculator input for loading subalgebras directly without recomputation. Subalgebras found so far: 119
Orbit sizes: A^165_1: n/a; A^85_1: n/a; A^84_1: n/a; A^45_1: 960; A^40_1: n/a; A^37_1: n/a; A^36_1: n/a; A^35_1: 480; A^21_1: n/a; A^20_1: 480; A^18_1: 960; A^13_1: n/a; A^12_1: n/a; A^11_1: 240; A^10_1: 80; A^10_1: 480; A^9_1: 240; A^8_1: 40; A^5_1: 32; A^4_1: 80; A^3_1: 80; A^2_1: 40; A^1_1: 10;
Current subalgebra chain length: 0
SetOutputFile("subalgebras_C^{1}_5");
LoadSemisimpleSubalgebras {}(AmbientDynkinType=C^{1}{}\left(5\right);CurrentChain=\left(\right);NumExploredTypes=\left(\right);NumExploredHs=\left(\right);Subalgebras=\left((DynkinType=A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right)\right)), (DynkinType=A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right)\right)), (DynkinType=A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 6 & 3 \end{pmatrix};generators=\left(g{}\left(-19\right)+g{}\left(-24\right), g{}\left(19\right)+g{}\left(24\right)\right)), (DynkinType=A^{4}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 4 \end{pmatrix};generators=\left(g{}\left(-24\right)+g{}\left(-16\right), g{}\left(24\right)+g{}\left(16\right)\right)), (DynkinType=A^{5}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 5 \end{pmatrix};generators=\left(g{}\left(-5\right)+g{}\left(-24\right)+g{}\left(-16\right), g{}\left(5\right)+g{}\left(24\right)+g{}\left(16\right)\right)), (DynkinType=A^{8}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4 \end{pmatrix};generators=\left(g{}\left(-17\right)+g{}\left(-11\right), 2 g{}\left(17\right)+2 g{}\left(11\right)\right)), (DynkinType=A^{9}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 10 & 10 & 5 \end{pmatrix};generators=\left(g{}\left(-19\right)+g{}\left(-17\right)+g{}\left(-11\right), g{}\left(19\right)+2 g{}\left(17\right)+2 g{}\left(11\right)\right)), (DynkinType=A^{10}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 10 & 12 & 6 \end{pmatrix};generators=\left(g{}\left(-11\right)+g{}\left(-17\right)+g{}\left(-16\right), 2 g{}\left(11\right)+2 g{}\left(17\right)+g{}\left(16\right)\right)), (DynkinType=A^{10}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right)\right)), (DynkinType=A^{11}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 10 & 10 & 5 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right)+g{}\left(-19\right), 3 g{}\left(1\right)+4 g{}\left(23\right)+g{}\left(19\right)\right)), (DynkinType=A^{12}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 10 & 12 & 6 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right)+g{}\left(-16\right), 3 g{}\left(1\right)+4 g{}\left(23\right)+g{}\left(16\right)\right)), (DynkinType=A^{13}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 10 & 12 & 7 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right)+g{}\left(-19\right)+g{}\left(-9\right), 3 g{}\left(1\right)+4 g{}\left(23\right)+g{}\left(19\right)+g{}\left(9\right)\right)), (DynkinType=A^{18}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 10 & 14 & 16 & 8 \end{pmatrix};generators=\left(g{}\left(-8\right)+g{}\left(-10\right)+g{}\left(-15\right)+g{}\left(-13\right), 2 g{}\left(8\right)+3 g{}\left(10\right)+2 g{}\left(15\right)+4 g{}\left(13\right)\right)), (DynkinType=A^{20}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 12 & 14 & 16 & 8 \end{pmatrix};generators=\left(g{}\left(-2\right)+g{}\left(-16\right)+g{}\left(-10\right), 3 g{}\left(2\right)+4 g{}\left(16\right)+3 g{}\left(10\right)\right)), (DynkinType=A^{21}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 12 & 14 & 16 & 9 \end{pmatrix};generators=\left(g{}\left(-2\right)+g{}\left(-5\right)+g{}\left(-16\right)+g{}\left(-10\right), 3 g{}\left(2\right)+g{}\left(5\right)+4 g{}\left(16\right)+3 g{}\left(10\right)\right)), (DynkinType=A^{35}{}\left(1\right);ElementsCartan=\begin{pmatrix}10 & 16 & 18 & 18 & 9 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-19\right), 5 g{}\left(1\right)+8 g{}\left(2\right)+9 g{}\left(19\right)\right)), (DynkinType=A^{36}{}\left(1\right);ElementsCartan=\begin{pmatrix}10 & 16 & 18 & 20 & 10 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-19\right)+g{}\left(-13\right), 5 g{}\left(1\right)+8 g{}\left(2\right)+9 g{}\left(19\right)+g{}\left(13\right)\right)), (DynkinType=A^{37}{}\left(1\right);ElementsCartan=\begin{pmatrix}10 & 16 & 18 & 20 & 11 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-19\right)+g{}\left(-9\right), 5 g{}\left(1\right)+8 g{}\left(2\right)+9 g{}\left(19\right)+g{}\left(9\right)\right)), (DynkinType=A^{40}{}\left(1\right);ElementsCartan=\begin{pmatrix}8 & 16 & 20 & 24 & 12 \end{pmatrix};generators=\left(g{}\left(-2\right)+g{}\left(-4\right)+g{}\left(-12\right)+g{}\left(-10\right), 4 g{}\left(2\right)+6 g{}\left(4\right)+6 g{}\left(12\right)+4 g{}\left(10\right)\right)), (DynkinType=A^{45}{}\left(1\right);ElementsCartan=\begin{pmatrix}10 & 16 & 22 & 24 & 13 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-3\right)+g{}\left(-11\right)+g{}\left(-9\right)+\frac{5}{2} g{}\left(-7\right), 5 g{}\left(1\right)+3 g{}\left(3\right)+g{}\left(5\right)+g{}\left(6\right)+g{}\left(7\right)-\frac{15}{2} g{}\left(8\right)+\frac{13}{2} g{}\left(9\right)+\frac{11}{2} g{}\left(11\right)+\frac{25}{4} g{}\left(13\right)\right)), (DynkinType=A^{84}{}\left(1\right);ElementsCartan=\begin{pmatrix}14 & 24 & 30 & 32 & 16 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-3\right)+g{}\left(-13\right), 7 g{}\left(1\right)+12 g{}\left(2\right)+15 g{}\left(3\right)+16 g{}\left(13\right)\right)), (DynkinType=A^{85}{}\left(1\right);ElementsCartan=\begin{pmatrix}14 & 24 & 30 & 32 & 17 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-3\right)+g{}\left(-5\right)+g{}\left(-13\right), 7 g{}\left(1\right)+12 g{}\left(2\right)+15 g{}\left(3\right)+g{}\left(5\right)+16 g{}\left(13\right)\right)), (DynkinType=A^{165}{}\left(1\right);ElementsCartan=\begin{pmatrix}18 & 32 & 42 & 48 & 25 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-3\right)+g{}\left(-4\right)+g{}\left(-5\right), 9 g{}\left(1\right)+16 g{}\left(2\right)+21 g{}\left(3\right)+24 g{}\left(4\right)+25 g{}\left(5\right)\right)), (DynkinType=2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ 0 & 2 & 2 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(-23\right), g{}\left(23\right)\right)), (DynkinType=A^{2}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & 0 & 2 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(-19\right), g{}\left(19\right)\right)), (DynkinType=2 A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & 0 & 2 & 4 & 2 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(-16\right), g{}\left(16\right)\right)), (DynkinType=A^{3}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 6 & 3\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-19\right)+g{}\left(-24\right), g{}\left(19\right)+g{}\left(24\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=A^{3}{}\left(1\right)+A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 6 & 3\\ 0 & 0 & 0 & 2 & 2 \end{pmatrix};generators=\left(g{}\left(-19\right)+g{}\left(-24\right), g{}\left(19\right)+g{}\left(24\right), g{}\left(-9\right), g{}\left(9\right)\right)), (DynkinType=A^{4}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 4\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right)+g{}\left(-16\right), g{}\left(24\right)+g{}\left(16\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=2 A^{4}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 4\\ 2 & 4 & 2 & 0 & 0 \end{pmatrix};generators=\left(g{}\left(-21\right)+g{}\left(-20\right), g{}\left(21\right)+g{}\left(20\right), g{}\left(-7\right)-g{}\left(-6\right), g{}\left(7\right)-g{}\left(6\right)\right)), (DynkinType=A^{5}{}\left(1\right)+A^{4}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 5\\ 2 & 4 & 4 & 2 & 0 \end{pmatrix};generators=\left(g{}\left(-19\right)+g{}\left(-17\right)+g{}\left(-18\right), g{}\left(19\right)+g{}\left(17\right)+g{}\left(18\right), g{}\left(-11\right)-g{}\left(-10\right), g{}\left(11\right)-g{}\left(10\right)\right)), (DynkinType=A^{8}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 0 & 0 & 2 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-17\right)+g{}\left(-11\right), 2 g{}\left(17\right)+2 g{}\left(11\right), g{}\left(-19\right), g{}\left(19\right)\right)), (DynkinType=A^{8}{}\left(1\right)+A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 0 & 0 & 2 & 4 & 2 \end{pmatrix};generators=\left(g{}\left(-17\right)+g{}\left(-11\right), 2 g{}\left(17\right)+2 g{}\left(11\right), g{}\left(-16\right), g{}\left(16\right)\right)), (DynkinType=A^{8}{}\left(1\right)+A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 2 & 0 & 2 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-6\right)+g{}\left(-21\right), 2 g{}\left(6\right)+2 g{}\left(21\right), g{}\left(-1\right)+g{}\left(-19\right), g{}\left(1\right)+g{}\left(19\right)\right)), (DynkinType=A^{8}{}\left(1\right)+A^{4}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 2 & 0 & 2 & 4 & 2 \end{pmatrix};generators=\left(g{}\left(-18\right)+g{}\left(-10\right), 2 g{}\left(18\right)+2 g{}\left(10\right), g{}\left(-1\right)+g{}\left(-13\right)+g{}\left(-19\right), g{}\left(1\right)+g{}\left(13\right)+g{}\left(19\right)\right)), (DynkinType=A^{8}{}\left(1\right)+A^{5}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 2 & 0 & 2 & 4 & 3 \end{pmatrix};generators=\left(g{}\left(-15\right)+g{}\left(-14\right), 2 g{}\left(15\right)+2 g{}\left(14\right), g{}\left(-1\right)+g{}\left(-5\right)+g{}\left(-16\right), g{}\left(1\right)+g{}\left(5\right)+g{}\left(16\right)\right)), (DynkinType=A^{9}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 10 & 10 & 5\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-19\right)+g{}\left(-17\right)+g{}\left(-11\right), g{}\left(19\right)+2 g{}\left(17\right)+2 g{}\left(11\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=A^{9}{}\left(1\right)+A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 10 & 10 & 5\\ 2 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-10\right)+g{}\left(-18\right)+g{}\left(-19\right), 2 g{}\left(10\right)+2 g{}\left(18\right)+g{}\left(19\right), g{}\left(-1\right)+g{}\left(-13\right), g{}\left(1\right)+g{}\left(13\right)\right)), (DynkinType=A^{9}{}\left(1\right)+A^{4}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 10 & 10 & 5\\ 2 & 0 & 0 & 2 & 2 \end{pmatrix};generators=\left(g{}\left(-19\right)+g{}\left(-14\right)+g{}\left(-15\right), g{}\left(19\right)+2 g{}\left(14\right)+2 g{}\left(15\right), g{}\left(-1\right)+g{}\left(-5\right)+g{}\left(-13\right), g{}\left(1\right)+g{}\left(5\right)+g{}\left(13\right)\right)), (DynkinType=A^{10}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 2 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-19\right), g{}\left(19\right)\right)), (DynkinType=A^{10}{}\left(1\right)+A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 2 & 4 & 2 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-16\right), g{}\left(16\right)\right)), (DynkinType=A^{10}{}\left(1\right)+A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 10 & 12 & 6\\ 2 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-16\right)+g{}\left(-14\right)+g{}\left(-15\right), g{}\left(16\right)+2 g{}\left(14\right)+2 g{}\left(15\right), g{}\left(-1\right)+g{}\left(-5\right), g{}\left(1\right)+g{}\left(5\right)\right)), (DynkinType=A^{10}{}\left(1\right)+A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 2 & 4 & 3 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-19\right)+g{}\left(-9\right), g{}\left(19\right)+g{}\left(9\right)\right)), (DynkinType=A^{10}{}\left(1\right)+A^{8}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 4 & 8 & 4 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-4\right)+g{}\left(-12\right), 2 g{}\left(4\right)+2 g{}\left(12\right)\right)), (DynkinType=2 A^{10}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 6 & 8 & 4 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-3\right)+g{}\left(-13\right), 3 g{}\left(3\right)+4 g{}\left(13\right)\right)), (DynkinType=A^{11}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 10 & 10 & 5\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right)+g{}\left(-19\right), 3 g{}\left(1\right)+4 g{}\left(23\right)+g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=A^{11}{}\left(1\right)+A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 10 & 10 & 5\\ 0 & 0 & 0 & 2 & 2 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right)+g{}\left(-19\right), 3 g{}\left(1\right)+4 g{}\left(23\right)+g{}\left(19\right), g{}\left(-9\right), g{}\left(9\right)\right)), (DynkinType=A^{11}{}\left(1\right)+A^{10}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 10 & 10 & 5\\ 0 & 0 & 0 & 6 & 4 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right)+g{}\left(-19\right), 3 g{}\left(1\right)+4 g{}\left(23\right)+g{}\left(19\right), g{}\left(-4\right)+g{}\left(-5\right), 3 g{}\left(4\right)+4 g{}\left(5\right)\right)), (DynkinType=A^{12}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 10 & 12 & 6\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right)+g{}\left(-16\right), 3 g{}\left(1\right)+4 g{}\left(23\right)+g{}\left(16\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{13}{}\left(1\right)+A^{8}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 10 & 12 & 7\\ 0 & 4 & 4 & 4 & 0 \end{pmatrix};generators=\left(g{}\left(-6\right)+g{}\left(-19\right)+g{}\left(-13\right)+g{}\left(-15\right), 3 g{}\left(6\right)+4 g{}\left(19\right)+g{}\left(13\right)+g{}\left(15\right), g{}\left(-7\right)-g{}\left(-4\right), 2 g{}\left(7\right)-2 g{}\left(4\right)\right)), (DynkinType=A^{18}{}\left(1\right)+A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 10 & 14 & 16 & 8\\ 0 & 2 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-10\right)+g{}\left(-11\right)+g{}\left(-12\right)+g{}\left(-13\right), 3 g{}\left(10\right)+2 g{}\left(11\right)+2 g{}\left(12\right)+4 g{}\left(13\right), g{}\left(-2\right)+g{}\left(-5\right), g{}\left(2\right)+g{}\left(5\right)\right)), (DynkinType=A^{20}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 12 & 14 & 16 & 8\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-2\right)+g{}\left(-16\right)+g{}\left(-10\right), 3 g{}\left(2\right)+4 g{}\left(16\right)+3 g{}\left(10\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{35}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}10 & 16 & 18 & 18 & 9\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-19\right), 5 g{}\left(1\right)+8 g{}\left(2\right)+9 g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=A^{35}{}\left(1\right)+A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}10 & 16 & 18 & 18 & 9\\ 0 & 0 & 0 & 2 & 2 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-19\right), 5 g{}\left(1\right)+8 g{}\left(2\right)+9 g{}\left(19\right), g{}\left(-9\right), g{}\left(9\right)\right)), (DynkinType=A^{35}{}\left(1\right)+A^{10}{}\left(1\right);ElementsCartan=\begin{pmatrix}10 & 16 & 18 & 18 & 9\\ 0 & 0 & 0 & 6 & 4 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-19\right), 5 g{}\left(1\right)+8 g{}\left(2\right)+9 g{}\left(19\right), g{}\left(-4\right)+g{}\left(-5\right), 3 g{}\left(4\right)+4 g{}\left(5\right)\right)), (DynkinType=A^{36}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}10 & 16 & 18 & 20 & 10\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-19\right)+g{}\left(-13\right), 5 g{}\left(1\right)+8 g{}\left(2\right)+9 g{}\left(19\right)+g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{40}{}\left(1\right)+A^{5}{}\left(1\right);ElementsCartan=\begin{pmatrix}8 & 16 & 20 & 24 & 12\\ 2 & 0 & 2 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-6\right)+g{}\left(-7\right)+g{}\left(-8\right)+g{}\left(-9\right), 4 g{}\left(6\right)+4 g{}\left(7\right)+6 g{}\left(8\right)+6 g{}\left(9\right), g{}\left(-1\right)+g{}\left(-3\right)+g{}\left(-5\right), g{}\left(1\right)+g{}\left(3\right)+g{}\left(5\right)\right)), (DynkinType=A^{84}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}14 & 24 & 30 & 32 & 16\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-3\right)+g{}\left(-13\right), 7 g{}\left(1\right)+12 g{}\left(2\right)+15 g{}\left(3\right)+16 g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=B^{1}{}\left(2\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right)\right)), (DynkinType=A^{2}{}\left(2\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right)\right)), (DynkinType=B^{2}{}\left(2\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ -2 & -4 & -2 & 0 & 0 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(10\right)+g{}\left(2\right), g{}\left(-10\right)+g{}\left(-2\right)\right)), (DynkinType=B^{4}{}\left(2\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 4\\ 0 & 0 & -4 & -8 & -4 \end{pmatrix};generators=\left(g{}\left(-18\right)+g{}\left(-22\right), g{}\left(18\right)+g{}\left(22\right), g{}\left(12\right)+g{}\left(4\right), 2 g{}\left(-12\right)+2 g{}\left(-4\right)\right)), (DynkinType=3 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ 0 & 2 & 2 & 2 & 1\\ 0 & 0 & 2 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(-23\right), g{}\left(23\right), g{}\left(-19\right), g{}\left(19\right)\right)), (DynkinType=A^{2}{}\left(1\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=2 A^{2}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & 0 & 2 & 4 & 2\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(-16\right), g{}\left(16\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{3}{}\left(1\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 6 & 3\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-19\right)+g{}\left(-24\right), g{}\left(19\right)+g{}\left(24\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=2 A^{4}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 4\\ 2 & 4 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-21\right)+g{}\left(-20\right), g{}\left(21\right)+g{}\left(20\right), g{}\left(-7\right)-g{}\left(-6\right), g{}\left(7\right)-g{}\left(6\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=3 A^{4}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 4\\ 2 & 4 & 2 & 0 & 0\\ 2 & 0 & 2 & 0 & 0 \end{pmatrix};generators=\left(g{}\left(-21\right)+g{}\left(-20\right), g{}\left(21\right)+g{}\left(20\right), g{}\left(-7\right)-g{}\left(-6\right), g{}\left(7\right)-g{}\left(6\right), -g{}\left(-1\right)+g{}\left(-3\right), -g{}\left(1\right)+g{}\left(3\right)\right)), (DynkinType=A^{5}{}\left(1\right)+2 A^{4}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 5\\ 2 & 4 & 4 & 2 & 0\\ 2 & 0 & 0 & 2 & 0 \end{pmatrix};generators=\left(g{}\left(-19\right)+g{}\left(-17\right)+g{}\left(-18\right), g{}\left(19\right)+g{}\left(17\right)+g{}\left(18\right), g{}\left(-11\right)-g{}\left(-10\right), g{}\left(11\right)-g{}\left(10\right), -g{}\left(-1\right)+g{}\left(-4\right), -g{}\left(1\right)+g{}\left(4\right)\right)), (DynkinType=A^{8}{}\left(1\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-17\right)+g{}\left(-11\right), 2 g{}\left(17\right)+2 g{}\left(11\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=A^{8}{}\left(1\right)+A^{3}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 2 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-6\right)+g{}\left(-21\right), 2 g{}\left(6\right)+2 g{}\left(21\right), g{}\left(-1\right)+g{}\left(-19\right), g{}\left(1\right)+g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=A^{8}{}\left(1\right)+A^{3}{}\left(1\right)+A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 2 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 2 \end{pmatrix};generators=\left(g{}\left(-6\right)+g{}\left(-21\right), 2 g{}\left(6\right)+2 g{}\left(21\right), g{}\left(-1\right)+g{}\left(-19\right), g{}\left(1\right)+g{}\left(19\right), g{}\left(-9\right), g{}\left(9\right)\right)), (DynkinType=A^{8}{}\left(1\right)+A^{4}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 2 & 0 & 2 & 4 & 2\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-18\right)+g{}\left(-10\right), 2 g{}\left(18\right)+2 g{}\left(10\right), g{}\left(-1\right)+g{}\left(-13\right)+g{}\left(-19\right), g{}\left(1\right)+g{}\left(13\right)+g{}\left(19\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{9}{}\left(1\right)+A^{3}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 10 & 10 & 5\\ 2 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-10\right)+g{}\left(-18\right)+g{}\left(-19\right), 2 g{}\left(10\right)+2 g{}\left(18\right)+g{}\left(19\right), g{}\left(-1\right)+g{}\left(-13\right), g{}\left(1\right)+g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{10}{}\left(1\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=A^{10}{}\left(1\right)+A^{2}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 2 & 4 & 2\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-16\right), g{}\left(16\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{10}{}\left(1\right)+A^{8}{}\left(1\right)+A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 4 & 8 & 4\\ 0 & 0 & 2 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-8\right)+g{}\left(-9\right), 2 g{}\left(8\right)+2 g{}\left(9\right), g{}\left(-3\right)+g{}\left(-5\right), g{}\left(3\right)+g{}\left(5\right)\right)), (DynkinType=2 A^{10}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 6 & 8 & 4\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-3\right)+g{}\left(-13\right), 3 g{}\left(3\right)+4 g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{11}{}\left(1\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 10 & 10 & 5\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right)+g{}\left(-19\right), 3 g{}\left(1\right)+4 g{}\left(23\right)+g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{35}{}\left(1\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}10 & 16 & 18 & 18 & 9\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-2\right)+g{}\left(-19\right), 5 g{}\left(1\right)+8 g{}\left(2\right)+9 g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-19\right), g{}\left(19\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 4 & 2 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-16\right), g{}\left(16\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 4 & 3 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-19\right)+g{}\left(-9\right), g{}\left(19\right)+g{}\left(9\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{8}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 4 & 8 & 4 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-4\right)+g{}\left(-12\right), 2 g{}\left(4\right)+2 g{}\left(12\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{10}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 6 & 8 & 4 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-3\right)+g{}\left(-13\right), 3 g{}\left(3\right)+4 g{}\left(13\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{11}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 6 & 8 & 5 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-3\right)+g{}\left(-5\right)+g{}\left(-13\right), 3 g{}\left(3\right)+g{}\left(5\right)+4 g{}\left(13\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{35}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 10 & 16 & 9 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-3\right)+g{}\left(-4\right)+g{}\left(-5\right), 5 g{}\left(3\right)+8 g{}\left(4\right)+9 g{}\left(5\right)\right)), (DynkinType=A^{2}{}\left(2\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=A^{2}{}\left(2\right)+A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 2 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(-9\right), g{}\left(9\right)\right)), (DynkinType=A^{2}{}\left(2\right)+A^{10}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 6 & 4 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(-4\right)+g{}\left(-5\right), 3 g{}\left(4\right)+4 g{}\left(5\right)\right)), (DynkinType=B^{2}{}\left(2\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ -2 & -4 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(10\right)+g{}\left(2\right), g{}\left(-10\right)+g{}\left(-2\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=B^{4}{}\left(2\right)+A^{5}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 4\\ 0 & 0 & -4 & -8 & -4\\ 2 & 0 & 2 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-20\right)+g{}\left(-21\right), g{}\left(20\right)+g{}\left(21\right), g{}\left(9\right)+g{}\left(8\right), 2 g{}\left(-9\right)+2 g{}\left(-8\right), g{}\left(-1\right)-g{}\left(-3\right)-g{}\left(-5\right), g{}\left(1\right)-g{}\left(3\right)-g{}\left(5\right)\right)), (DynkinType=C^{1}{}\left(3\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & -2 & -1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(19\right), g{}\left(-19\right)\right)), (DynkinType=A^{2}{}\left(3\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(3\right), g{}\left(-3\right)\right)), (DynkinType=4 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ 0 & 2 & 2 & 2 & 1\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(-23\right), g{}\left(23\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=A^{2}{}\left(1\right)+3 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=3 A^{4}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 6 & 8 & 4\\ 2 & 4 & 2 & 0 & 0\\ 2 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-21\right)+g{}\left(-20\right), g{}\left(21\right)+g{}\left(20\right), g{}\left(-7\right)-g{}\left(-6\right), g{}\left(7\right)-g{}\left(6\right), -g{}\left(-1\right)+g{}\left(-3\right), -g{}\left(1\right)+g{}\left(3\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{8}{}\left(1\right)+A^{3}{}\left(1\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}4 & 8 & 8 & 8 & 4\\ 2 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-6\right)+g{}\left(-21\right), 2 g{}\left(6\right)+2 g{}\left(21\right), g{}\left(-1\right)+g{}\left(-19\right), g{}\left(1\right)+g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{10}{}\left(1\right)+3 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}6 & 8 & 8 & 8 & 4\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-1\right)+g{}\left(-23\right), 3 g{}\left(1\right)+4 g{}\left(23\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=B^{1}{}\left(2\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{2}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 4 & 2\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-16\right), g{}\left(16\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{8}{}\left(1\right)+A^{3}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 4 & 8 & 4\\ 0 & 0 & 2 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-8\right)+g{}\left(-9\right), 2 g{}\left(8\right)+2 g{}\left(9\right), g{}\left(-3\right)+g{}\left(-5\right), g{}\left(3\right)+g{}\left(5\right)\right)), (DynkinType=B^{1}{}\left(2\right)+A^{10}{}\left(1\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 6 & 8 & 4\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-3\right)+g{}\left(-13\right), 3 g{}\left(3\right)+4 g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=2 B^{1}{}\left(2\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & -2 & 0 & 0 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(3\right), g{}\left(-3\right)\right)), (DynkinType=A^{2}{}\left(2\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=A^{2}{}\left(2\right)+B^{1}{}\left(2\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & -2 & 0 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(4\right), g{}\left(-4\right)\right)), (DynkinType=C^{1}{}\left(3\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & -2 & -1\\ 0 & 0 & 0 & 2 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(19\right), g{}\left(-19\right), g{}\left(-13\right), g{}\left(13\right)\right)), (DynkinType=C^{1}{}\left(3\right)+A^{2}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & -2 & -1\\ 0 & 0 & 0 & 2 & 2 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(19\right), g{}\left(-19\right), g{}\left(-9\right), g{}\left(9\right)\right)), (DynkinType=C^{1}{}\left(3\right)+A^{10}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & -2 & -1\\ 0 & 0 & 0 & 6 & 4 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(19\right), g{}\left(-19\right), g{}\left(-4\right)+g{}\left(-5\right), 3 g{}\left(4\right)+4 g{}\left(5\right)\right)), (DynkinType=A^{2}{}\left(3\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(3\right), g{}\left(-3\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=C^{1}{}\left(4\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & -2 & -1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(3\right), g{}\left(-3\right), g{}\left(13\right), g{}\left(-13\right)\right)), (DynkinType=A^{2}{}\left(4\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(3\right), g{}\left(-3\right), g{}\left(4\right), g{}\left(-4\right)\right)), (DynkinType=5 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ 0 & 2 & 2 & 2 & 1\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(-23\right), g{}\left(23\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=B^{1}{}\left(2\right)+3 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=2 B^{1}{}\left(2\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 2 & 2 & 2 & 1\\ -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 2 & 1\\ 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-25\right), g{}\left(25\right), g{}\left(1\right), g{}\left(-1\right), g{}\left(-19\right), g{}\left(19\right), g{}\left(3\right), g{}\left(-3\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=C^{1}{}\left(3\right)+2 A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & -2 & -1\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(19\right), g{}\left(-19\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=C^{1}{}\left(3\right)+B^{1}{}\left(2\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & -2 & -1\\ 0 & 0 & 0 & 2 & 1\\ 0 & 0 & 0 & -2 & 0 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(19\right), g{}\left(-19\right), g{}\left(-13\right), g{}\left(13\right), g{}\left(4\right), g{}\left(-4\right)\right)), (DynkinType=C^{1}{}\left(4\right)+A^{1}{}\left(1\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & -2 & -1\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(3\right), g{}\left(-3\right), g{}\left(13\right), g{}\left(-13\right), g{}\left(-5\right), g{}\left(5\right)\right)), (DynkinType=C^{1}{}\left(5\right);ElementsCartan=\begin{pmatrix}2 & 4 & 4 & 4 & 2\\ 0 & -2 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & -1 \end{pmatrix};generators=\left(g{}\left(-24\right), g{}\left(24\right), g{}\left(2\right), g{}\left(-2\right), g{}\left(3\right), g{}\left(-3\right), g{}\left(4\right), g{}\left(-4\right), g{}\left(5\right), g{}\left(-5\right)\right))\right))